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PREFACE TO SECOND EDITION. 

The favorable reception accorded to this book upon its 
original appearance and its extensive use for nine years in 
many colleges has demonstrated that the original plan was 
well-suited for student use. That plan has been adhered 
to in preparing the present edition. The chaotic condition 
of the world as a consequence of the war reflects itself in 
the traction enterprises at present, but it is safe to say that 
they will persist and that the problems of their reconstruc- 
tion will fall upon the engineers. As a fundamental prep- 
aration for this kind of work it is believed the present 
edition will prove serviceable. 

Polytechnic Institute, Brooklyn, N. F., 
February /, ig20. 



7 



0' 



y^^'-'' 



PREFACE TO FIRST EDITION. 



The ultimate purpose of nearly all the professional 
efforts of an engineer is the attainment of efficiency in the 
utilization of labor, capital, and energy. To attain the 
highest efficiency in the construction and the subsequent 
operation of a complete installation requires a knowledge 
of the facts and a familiarity with the laws pertaining to 
these three factors. Decisions as to the selection of the 
type and the dimensions of an element, often attributed 
to the exercise of good judgment, are generally the specific 
results of the correct application of laws to all pertinent 
facts. 

The number of facts to be considered in determining the 
final elements of a complete electric traction system is 
enormous. As a consequence students and young engineers 
become bewildered and are unable to discriminate as to the 
pertinency or necessity of specific details. To meet this 
condition the present text has been prepared, it being 
believed that no other single published book meets it. 

The book attempts to present a perspective view of the 
design of a complete railway installation, from the cars 
to the power-station, to indicate the nature and sequence of 
the various entailed problems, and to suggest or illustrate 
methods for their solution. 

In preparing the text the determination of what to omit 
has involved nearly as much effort as of what to include. 

vii 



vili PREFACE 

A descriptive treatment of specific forms of structures has 
been avoided. On the other hand, a number of numerical 
illustrations of the calculation of economic magnitudes has 
been given. Again, the inevitable future extensive use of 
hyperbolic functions has claimed for them a brief but 
comprehensive exposition and their utility is demonstrated 
in connection with calculations relating to electric-wave 
propagation. 

Appreciation is hereby expressed of the services of Mr. 
G. I. Rhodes in making helpful suggestions and in reading 
the proofs of the sections on economic determinations. 

Polytechnic Institute, Brooklyn, N. Y. 
May I, 191 1. 



CONTENTS. 



CHAPTER I. 

Determination of the Number and Size of Cars for an 
Urban Road. 

ART. PAGE 

1. The Engineer's Problem i 

2. Types of Service i 

3. Length of Track 2 

4. Receipts 4 

5. Number of Cars 4 

6. Size of Cars 8 

7. Trains 12 

Problems 14 

CHAPTER II. 

Tractive Effort Required for Car Propulsion. 

8. Train Resistance 15 

9. Grades 19 

10. Curves 20 

11. Acceleration 22 

12. Braking 23 

Problems 25 

CHAPTER III. 
Types and Performance Curves of Motors. 

13. Traction Motors 26 

14. Direct-current Motors 27 

15. Alternating-current Motors 28 

16. Methods of Drive 41 

17. Motor Curves 44 

Problems 49 

ix 



X CONTENTS. 

CHAPTER IV. 

Speed Curves. 

ART. PAGE 

i8. Motor Limitations 50 

19. Motor Capacity 51 

20. Speed 51 

21. Typical Speed Curves 52 

22. Data for Plotting Speed Curves 53 

23. Plotting Speed Curves 56 

24. Numerical Example 59 

25. Distance Curves 66 

26. Speed Curve Plotting with Grades and Curves 67 

Problems 73 

CHAPTER V. 

Railway Motor Control. 

27. Direct-current Control 74 

28. Rheostatic Method 74 

29. Series-parallel Method 75 

30. Starting Resistances 78 

31. Numerical Example 87 

32. Field Control 89 

33. Alternating-current Control 90 

34. Compensators 93 

35. Induction Motor Control 96 

36. Controllers 103 

Problems no 

CHAPTER VI. 

Energy Consumption. 

37. Current Curves in 

38. Average and Effective Currents 112 

39. Numerical Example 113 

40. Effective Motor Current for a Trip 116 

41. Voltage Curve 118 

42. Motor Heating 118 

43. Energy for Direct-current Propulsion 120 

44. Energy for Alternating-Current Propulsion 121 



CONTENTS. XI 

ART. PAGE 

45. Effect of Operating Conditions on Energy Consumption 124 

46. Gear Ratio 130 

Problems 132 

CHAPTER VII. 

The Distributing System. 

47. Classification of Conductors 133 

48. Contact Conductors 134 

49. Branches 139 

50. Collecting Devices 140 

51. Supplementary Conductors 142 

52. Graphic Time-table 147 

53. Feeders 151 

54. Track Rails , 155 

55. Negative Track Feeders 157 

56. Electrolytic Surveys. . ' 161 

57. Alternating-current Distribution 164 

Problems 164 

CHAPTER VIII. 

Substations. 

58. Types of Substations 166 

59. Direct Currents Received and Delivered 166 

60. Alternating Currents Received and Delivered 168 

61. Alternating Currents Received and Direct Currents Delivered . . . 169 

62. Location of Substations 175 

63. Numerical Illustration 186 

64. Auxiliary Storage Batteries 188 

65. Arrangement of Apparatus 189 

66. Portable Substations 193 

Problems 197 

CHAPTER IX. 

Transmission Lines. 

67. Location of the Transmission Line 199 

68. Number of Phases 201 

69. Frequency 203 

70. Economic Voltage 205 

71. Numerical Illustration 2H 



xii CONTENTS. 

ART. PAGE 

^2, Separation of Conductors 213 

73. Resistance of Conductors 220 

74. Line Inductance 222 

75. Hyperbolic Functions 224 

76. Line Capacity 230 

77. Equations of Wave Propagation along Wires 235 

78. Attenuation and Wave-length CoeflScients 238 

79. Current and Voltage Distribution on Lines 240 

80. Regulation 243 

81. Numerical Illustration 244 

82. Corona Loss 247 

^Z' Lightning 251 

84. Protection from Lightning 254 

Problems 258 

CHx\PTER X. 

Power Stations. 

85. Station Load Curves 259 

86. Selection of Generators 261 

%']. Types of Prime Movers 263 

88. Power Station Costs 264 

Steam Stations. 

89. Engines and Turbines 265 

90. Condensers 267 

91. Boilers 270 

92. Feed-water Heaters 272 

93. Chimneys or Stacks 272 

94. Buildings 274 

95. Arrangement of Apparatus 275 

96. Cost of Steam Stations 280 

97. Operating Expenses 280 

Hydraulic Stations, 

98. Turbines 281 

99. Water-power Development 288 

100. Cost of Development 293 

loi. Depreciation and Obsolescence 297 

102. Relative Operating Expenses 299 

103. Costs per Kilowatt-hour 299 

Problems 301 



ELECTRIC TRACTION AND TRANSMISSION 
ENGINEERING. 

CHAPTER I. 

DETERMINATION OF THE NUMBER AND SIZE OF CARS 
FOR AN URBAN ROAD. 

1. The Engineer's Problem. — The problem of the 
electric railway engineer is the determination of the car 
equipment required to yield a proposed service, the char- 
acteristics of the low-potential distribution system, the 
location and capacity of the substation equipment, the 
characteristics of the high-tension transmission line, and 
finally the capacity of the main generating station. His 
report should include cost estimates of the various items 
of the electric railway system, probable operating expenses 
and approximate gross income on the investment. 

2. Types of Service. — The object of a railway is the 
transportation of passengers or freight between any points 
on the road in accordance with a schedule which is pre- 
pared to accommodate the traffic most economically and 
to lead to a sufficient income on the original investment 
to the operating company. The probable location of a 
proposed electric railway is governed by purely local con- 
ditions, such as density of population, future growth of 
the community, and topography of the land. An approxi- 



2 TRACTION AND TRANSMISSION. 

mate estimate of the length of a proposed railway and its 
subsequent income, as well as the determination of the 
number and size of the cars or trains, may be obtained 
from government reports and other statistical sources. 

-Electric railway undertakings are of three kinds, — new 
roads, extensions to existing railways, and electrifications 
of present steam railroads. Of these, the former will first 
be considered. A new electric railway undertaking may 
relate to an urban, suburban, or interurban installation. 
Frequently a single system will include all of these types 
of service. 



3. Length of Track. — For a new urban street railway 
the economically feasible length of road will depend largely 
upon the population. Thus, curve i of Fig. i shows the 
number of miles of track per 1000 of population for various 
population centers. This curve represents the data of the 
following table showing the relation of trackage and traffic 
to population in groups of urban centers; it is taken from 
the Census Report on Electric Railways for 1902. The 
figures refer to single track, and for a double-track road the 
length of track is twice the length of the road. 



Total population 
served 

Number of miles of 
track 

Miles of track per 
1000 of population 

Number of passen- 
gers 

Number of rides 
per inhabitant . . . 



All centers 
over 500,000 
population. 



10,274,470 
4,998.89 
.49 
2,456,542,270 
239.1 



All centers 

between 100,000 

and 500,000 

population. 



5,380,647 

.66 
994,327,853 
184.7 



Twenty-nine 

selected centers 

between 25,000 

and 100,000 

population. 



1,258,615 

951-93 
.76 
135,842,312 
107.9 



Forty-six 
selected 
centers of 
less than 

25,000 
population. 



718,254 

485-95 
.68 
49,179,495 
68.5 



NUMBER AND SIZE OF CARS FOR URBAN ROAD. 



The present population is, however, not the value to be 
considered in determining the track factor ^ r, from this 
curve, but instead the population at some future time, 
this time depending upon the probable duration of the 



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MILLIONS 

AVERAGE POPULATION. 

Fig. I. 



period of construction, the depreciation, and later pro- 
spective developments in electric traction. The popula- 
tion, iV, at some future time may be estimated from the 
past growth of the community. Thus, a curve of popula- 
tion for the last one hundred years might be drawn and 



4 TRACTION AND TRANSMISSION. 

extended, or a percentage increase of population may be 
assumed. A population value corresponding to a time 
ten years later offers a reasonable working basis. Then 
the number of miles of track, Z, to be installed can be 
expressed as 

L = miles. 

lOOO 

4. Receipts. — In the foregoing table is also given the 
annual number of rides per inhabitant for various popu- 
lation centers, the data showing that passenger traffic is 
comparatively greater in the larger cities. The riding 
habit of people increases from year to year as the com- 
munity grows, as its business, family and social life be- 
comes more complex, and as its facilities for intercommun- 
ication improve. Curve 2, Fig. i, shows the number of 
yearly passengers per inhabitant, or what may be termed 
the passenger factor , 7. Then the number of passengers 
per year can be written as 

Yearly passengers = Ny, 

The annual receipts, in dollars, R, of a traction company 
are evidently the product of the total yearly passengers 
into the fare, /, in dollars, or 

R =iV^7/ dollars. 
In this country the usual urban fare prior to the war was 
five cents regardless of the distance traveled. For inter- 
urban roads the fare depends upon the distance traveled, 
varying from one to three cents per mile. 

5. Number of Cars. — The determination of the num- 
ber of cars to install may be made by the aid of tables 
which show the income and operating expenses per car 



NUMBER AND SIZE OF CARS FOR URBAN ROAD. 



5 



mile of a number of electric railways. The following table 
compiled by H. M. Beardsley gives such data for some 
electric railways in New York State for 1905. Herefrom 
the average income per car mile is 21.56 cents. 



Company. 



Albany & Hudson 

United Traction Co. of Albany .... 

Auburn and Syracuse Co. 

Binghamton Ry. Co 

International Tr. Co. of Buffalo . . . 

Rochester & Eastern 

Cortland Traction Co 

E. W., L. & R.R. Co., Elmira 

City Ry., L. & R. Co., Fishkill. . . . 

Dunkirk & Fredonia 

Hudson Valley Ry. Co., Glens Falls. 
Hornell Elec. Ry., Hornellsville. . . 

Ithaca St. Ry. Co., Ithaca 

King. Consol. R.R. Co., Kingston.. 
Orange County Trac. Co., New- 
burgh 

Ogdensburg St. Ry. Co 

I. C. & R. S. Ry. Co., Oneonta 



Income from 


Income per 


operation. 


car mile. 




Cents. 


$200,671.65 


28.50 


1,714,848.82 


22.35 


268,507.78 


25.12 


258,819.85 


20.14 


3,694,339.01 


25.16 


212,668.51 


27.88 


49,139-86 


22.95 


192,921.47 


16.06 


41,474-56 


24.17 


44,457-88 


26.92 


499,148.09 


25.89 


16,919.70 


9-30 


91,817.90 


23.21 


123,632.92 


23.08 


119,270.85 


20.04 


27,240.09 


9.78 


103,862.05 


15-97 



Total 
expense per 
car mile. 



Cents. 
24.30 

15-35 
16.24 
11.23 
14.90 
21.36 
16.06 
II .61 
16.34 
22.57 

18.13 

9.06 

17.87 

14.57 

15-39 

7.86 

13.82 



The following table presents information compiled by 
G. H. Davis and furnished by sixteen electric railway 
companies which represent both geographically and politi- 
cally nearly all sections of the United States and all con- 
ditions of operation. The values given are for the year 
1910; the average passenger earnings per car mile being 
27.31 cents. 

The growth of traction earnings in the larger American 
cities, together with the corresponding operating expenses 
on a car mileage basis are shown in Fig. 2, which was pre- 
pared by B. J. Arnold. It will be noted, for instance, that 

in Brooklyn the earnings per car mile (average for street 
2 



TRACTION AND TRANSMISSION. 



d 


ill 

Oh 




•11 

•o.S 

to 03 


p. 

173 oS 

u 




Average revenue 
per passenger, 
including trans- 
fer, in cents. 


Length of longest 
ride possible for 
one fare in miles. 


I 






485.2 
585-0 
208.2 
136.0 
IOI.7 
306.6 
1397 

no. 4 

86.8 
186.0 
129.4 

58.0 
1330 

41.6 
627.6 

330 


53,362,500 

37,537,433 


27.80 
28.77 


*3-i 
4-34 

*3-3 
3.12 
3.62 
3.89 

*3-3 
4.09 
4. 10 
4.09 
4.90 

4-94 
4.08 
*4.2 

4.15 
4.07 


20 


2 

3 
4 
5 
6 

7 
8 

9 

lO 

II 


533»905 
465,786 

373.740 

347,469 
516,152 
233,650 

131,105 
129,867 
155,000 

88,926 

51,521 
132,685 

36,346 
1,549,008 

46,000 


1,018,463 


14.4 
13-5 
12. 1 
14.6 

17.7 
12.6 
16. 1 

^■3 
18.0 
14.0 

7.S 
13.6 

9-5 
195 

8.8 


403,740 
512,886 
516,152 
234,650 
151,105 
216,867 
185,000 


13,812,813 
*i5,377,ooo 

24,229,010 
9,346,183 
6,895,421 
4,068,502 
9,538,867 


27.42 
31-50 
30.75 
28.86 
26.14 
28.70 

23-84 
26.14 


12 


60,521 

140,000 

71,346 

1,993,400 

46,500 




13 
14 
15 
i6 


6,194,583 

2,045,703 

70,943,404 

1,790,722 


26.32 
23.29 

25-34 
27.42 



* Estimated. 

and elevated railway service) increased from 24 cents in 
1902 to 29 cents in 1906 and then decreased to 26.8 cents 
in 1910. 

The total number of annual car miles to be operated is 
equal to the annual receipts divided by the annual income 
per car mile Rem', this result, when divided by 365 days 
and the daily number of hours of operation, h, gives the 
number of car miles to be operated per hour. If this be 
divided by the schedule speed, F, in miles per hour includ- 
ing stops, there results the number of cars required for the 
service. The schedule speed is limited by city ordinance 
in many cities to 12 miles per hour or less. The smallest 
number, v, of cars required then, may be expressed as 

= ^ = ^7/ 

""' 2,65hVKm 3^5hVReJ 



NUMBER AND SIZE OF CARS FOR URBAN ROAD. 7 





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8 TRACTION AND TRANSMISSION. 

6. Size of Cars. — The number of passengers carried 
per year divided by 365 and the number of cars in service 
gives the average number of passengers conveyed by each 
car per day. The number of trips per day made by each 
car is found by multiplying the schedule speed by the 
number of hours the car operates daily and dividing by the 
length of the line. The average number of passengers per 
trip is therefore ^ ^ ^yL _ NtR,^ 



365 vVh 1000/ 

When several lines are operated in the same district or 
city, the second member of this equation applies to each 
line of track-length L miles. With a single line the last 
member is applicable. 

The number of passengers riding in a car at different 
times varies widely, and it would be poor economy to em- 
ploy cars or trains of such size as to permit the average num- 
ber of passengers per trip, as obtained from the foregoing 
expression, to be seated at one time. Not all of these passen- 
gers ride the full length of the road, and again, others may 
stand. In a specific case information should be obtained, 
from records concerning similar cases, as to the average 
length of rides by passengers. Available data indicate that 
the average passenger ride, r, is from 2 miles to 4.5 miles. 

The length of track divided by the average length of 
ride determines the number of times that the car is refilled 
each trip. The average number of passengers per trip 
divided by this number gives the passenger capacity of a 

car as C = — - ^^^ 



L 365 vVh 

an expression which assumes uniform traffic conditions. 
With due consideration for the provision of additional 



NUMBER AND SIZE OF CARS FOR URBAN ROAD. 9 




Fig. 3. 




Fig- 4. 



seats for the accommodation of passengers during the rush 
hours, the seating capacity of the car is thus determined. 

CKmatic conditions and hmitations as to the total amount 
of roUing stock determine the characteristics of car-body 



lO 



TRACTION AND TRANSMISSION. 



construction as to whether it shall be open, closed, con- 
vertible, semiconvertible, double-decked, or combination 
open and closed. Figs. 3, 4 and 5 show the character- 




istic forms of construction of convertible, ^'Narragansett'* 
open, and semiconvertible interurban cars respectively. 

Fig. 6 (top) shows a pay-as-you-enter car which is being 
extensively adopted for congested urban traffic because it 
facilitates comfort, ingress and egress of passengers, and 
collection and conservation of fares. In non-congested 
traffic districts ''one-man" cars. Fig. 6 (bottom), are being 
used, wherewith the motorman is the only attendant; 
naturally the cost of operation is reduced. 

The arrangement of seats, as to whether they shall be 
transverse, longitudinal, or partly both, is dictated by the 
type of service to be rendered. Transverse seats are far 
more comfortable for seated passengers and are essential 
in long-haul service. Longitudinal seats greatly facilitate 
ingress and egress of passengers, give greater comfort to 
standing passengers, and as a rule permit of a greater ratio 
of standing to seated passengers. In urban and frequent- 



NUMBER AND SIZE OF CARS FOR URBAN ROAD. 



II 



stop service facility of ingress and egress is of paramount 
importance in order that a high schedule speed may be 
maintained. During the morning and evening rush hours 
the number of standing passengers frequently equals that 
of those seated. 




^E^^^^^^^^S.. 



""^^^^^^^ ^^^^^L 




Fig. 6. 



12 TRACTION AND TRANSMISSION. 

The weights of car bodies are always much greater than 
might be desired, but are necessitated in order to give 
adequate strength to withstand the rough usage of ordi- 
nary service and to give some insurance against collapse 
in case of collision. As will appear later, the first cost and 
expense of operation are dependent upon the total weight. 
The weight of passengers seldom reaches one-quarter the 
total weight. It is evidently desirable to reduce the weight 
of cars to a minimum consistent with adequate strength. 

The total weights of closed and semiconvertible cars of re- 
cent design are usually between 90 and 130 poundsper square 
foot of floor area, considering the floor area as the product 
of the length over bumpers by the width over belt rails. 

An analysis of the possible saving incident to the use of 
light cars in a group of street railway properties, having 
for 1 910 gross earnings of approximately $5,700,000, shows 
that of the 92.33 per cent of such earnings expended for 
all purposes, excluding dividends, including operating ex- 
penses 54.47 per cent, interest 24.74 per cent, taxes 7.12 
per cent, depreciation 6 per cent, only 53.08 per cent is 
influenced by car weight or Hve weight transported. Of 
this the items particularly affected are cost of power, car 
and track repairs, interest and depreciation, which in the 
aggregate do not generally exceed 15 per cent of the gross 
earnings. 

Having decided upon the seating capacity of the car, its 
size and weight may be determined from the following 
table. The average weight of a passenger may be taken 
as 140 pounds. The weights of trucks as given include 
the weights of motors except where starred. 



NUMBER AND SIZE OF CARS FOR URBAN ROAD. 13 

7. Trains. — In most of the large cities in the United 
States the traffic is so dense that the capacity of the largest 
car which it is practicable to employ is inadequate to meet 
conditions. It is customary therefore to operate trains of 
two or more cars. On surface roads a motor car connected 
with a trailer is often used and in southern cities it becomes 
possible to thus separate the colored and white people by 
assigning the trailer to the former passengers. 

In subways and on elevated roads lo-car trains are often 
operated. On the New York Municipal Railway each car 
accommodates 78 seated and 72 standing passengers, allow- 
ing 5 square feet of floor space to each standing individual. 
With cars of such size a number of doors should be provided 
so that the duration of stops will not be excessive. The 
cost of operation per car mile with trains is materially 
reduced by the use of side doors having remote-control 

CAR DATA. 



Type. 

Closed cars: 

Single truck 

Single truck 

Single truck 

Single motor 

Double truck 

Manhattan Elev. . . . 

I.R. T. Co. (steel). 

N.Y. C. (steel).... 
Open cars: 

8-bench 

lo-bench 

i2-bench 

14-bench 

Semiconvertible cars: 

Single truck 

Single truck. 

Double truck 

Double truck 



Length of 
body. 


Seating 
capacity. 


16' 


22 


18' 

20' 8'' 


24 
32 


28' 


38 


30' 8'' 

42' 

44' 


44 
58 
60 


50' 


70 


15' 8- 


32 


21' 

30' 2" 


50 
60 


30' 2" 


70 


18' 

20' 8" 


24 
32 


28' 
30' 8" 


40 
44 



Weight of 
body, 
pounds. 



6,000 

13.750 
11,310 
26,725 

22,000 

56,300 
85,100 

6,375 
13.340 
15.250 
20,300 

6,640 
10,240 
15,120 
19,500 



Weight of 
trucks, 
pounds. 



4,600* 

4,825 

5.125 

7.050 

14,500 

15,000* 

21,000* 

21,000* 

5.150 

5.925 

11,250 

7.550 

4,900 

5,100 

10,450 

10,800 



14 TRACTION AND TRANSMISSION. 

automatic door-opening devices, which practice reduces the 
number of employees to one attendant per car. 

On suburban sections of electric railways the schedule 
speed is most frequently from 15 to 20 miles per hour and 
on interurban sections from 25 to 3.5 miles per hour. The 
highest schedule speed at present for limited interurban 
service is 55 miles per hour on a 36-mile run. At high 
speeds the energy consumption per mile per ton of car 
weight is much greater for a single car than for a train of 
several cars, and consequently economical interurban opera- 
tion dictates the employment of trains of several units in- 
stead of single cars. It is interesting to note that the traffic 
on an interurban railway is furnished principally by the 
inhabitants of the towns, the rural districts supplying only 
from about 20 to 30% of the total traffic. 

PROBLEMS. 

1. How many cars, accommodating at a maximum 80 passengers each, 
should be used for a proposed electric railway for a city of the size indicated 
below? The schedule speed is specified at 10 miles per hour over three 
parallel lines of equal length, the period of operation to extend over the entire 
day. Take 4 miles as the average passenger ride, and assume the traflBc 
at rush-hours to be five times the average traflftc and to endure for about 
two hours morning and evening. The past growth of this city is indicated 
below : 

1850 2,000 inhabitants 

i860 4,000 " 

1870 8,000 " 

1880 17,000 " 

1890 42,000 " 

1900 90,000 " 

1910 200,000 " 

1920 350,000 ** 

2. Plot a curve showing the relation which should exist between the 
population of the city just referred to in former years, and the number 
of cars necessarv at those times. 



TRACTIVE EFFORT REQUIRED FOR CAR PROPULSION. 15 



CHAPTER II. 
TRACTIVE EFFORT REQUIRED FOR CAR PROPULSION. 

8. Train Resistance. — The determination of motor 
capacity for a proposed service involves a knowledge of 
the tractive effort to be exerted to produce the specified or 
assumed acceleration against the resistances offered by 
windage, friction, grades and curves, and also information 
about the performance of various sized motors such as is 
usually embodied in motor characteristic curves supplied 
by the manufacturers. The tractive effort, or force exerted 
at the rim of the car wheels, required to propel a car at 
constant speed on a straight level track is only that neces- 
sary to neutralize at that speed the resistance offered to 
car movement by bearing friction, rolling friction and 
flange friction on the track, and wind pressure; these 
resistances are considered under the single term train 
resistance. Many empirical formulae based upon experi- 
mental data have been proposed for use in estimating train 
resistance. A consideration of the various components of 
train resistance mentioned above will lead to the formu- 
lation of a fairly reliable expression therefor. 

Bearing friction, resulting from the sliding of the sur- 
faces of the axles over those of the journals, follows the 
ordinary laws of sliding friction. It depends upon the 
pressure between the surfaces, and increases slightly with 
speed. Rolling friction is due to deformation of the rails 
and wheel rims where they come in contact, and to un- 



l6 TRACTION AND TRANSMISSION. 

evennesses in the surface of the track. The energy con- 
sumed in overcoming rolHng friction is theoretically pro- 
portional to the weight on the track and to the distance 
covered. The force required to overcome it should there- 
fore be constant. It is^ however^ generally assumed to 
increase slightly with the velocity of the train. Experi- 
mental data thus far obtained warrant the following 
expression for the tractive effort necessary to overcome 
bearing and rolHng friction: 

R' = k + KV, 

where R^ is expressed in pounds tractive effort per ton of 
car weight, V is the speed in miles per hour, and k and K 
are constants. The value of k, since it depends upon the 
weight concentrated on the bearings, may be expressed in 
terms of train weight, W, in tons, and the expression 

Vw 
gives results agreeing well with experimental values, the 
minimum value of k being limited to 3.5. Values of K 
obtained experimentally vary from 0.03 to 0.07 depending 
upon track conditions and type of equipment, the lower 
values being the more representative. For light equipment 
and poor conditions of track the use of higher values is 
desirable. The resulting expression for bearing and roll- 
ing friction may then be written simply as 

50 V 
i?' = ~-^= H pounds per ton. 

The principal component of train resistance at high 
speeds is the wind pressure on the moving car. Wind 
pressure varies approximately as the square of the car 



TRACTIVE EFFORT REQUIRED FOR CAR PROPULSION. 1 7 

velocity, as shown by numerous experiments. Therefore 

an expression for head-end wind resistance takes the form 

R' = k'SV^ pounds, 

where S is the car cross section in square feet and k^ is a 
constant denoting the wind pressure per square foot at 
unit speed, the value of which depends upon the shape 
of the car end. For cars with perfectly flat ends its value 
would be about 0.004 ^.nd for cars of the pointed-nose 
design k^ is as low as 0.0015, whereas for city and suburban 
cars of the usual types and for the modern electric locomo- 
tives a value of 0.0025 ^^7 be taken with propriety. The 
wind pressure thus far considered is that on the car end, 
but there is also air resistance at the sides of the car or 
cars, which effect is particularly prominent in trains of 
several cars. There it becomes necessary to introduce a 
factor which takes care of this skin friction along the 
surface of succeeding cars, and it is usual to add 10 % of 
the head-end resistance as just obtained for each car follow- 
ing the first. Then, if n be the number of cars in the train, 
the tractive effort in pounds per ton of train weight is 

50 V sv r n - il 

Ri = —^ + — ^ + ^Wr I + "TTT" pounds per ton, 

a formula which combines the various expressions of the 
components of train resistance. Car cross sections may 
be taken as follows: 90 sq. ft. for 20-ton cars, 100 sq. ft. 
for 30-ton cars, no sq. ft. for 40-ton cars, and 120 sq. ft. 
for heavier cars. 

* A. H. Armstrong uses 33 and 500 as average values of the respective 
constants of the second and third terms in his formula. 



i8 



TRACTION AND TRANSMISSION. 



Fig. 7 shows by curves the dependence of train resistance 
of single cars upon speed and weight of car as determined 
by the foregoing formula. 

As an illustration, determine the tractive effort per ton 
exerted by an electric train of one or more cars when run- 



60 



z 50 
o 

C£. 
Mi 
Q. 

CO 40 
o 

z 

O 
Q. 

-30 

UJ 

o 

z 
< 

I- 

co 
w20 

hi 
DC 



< 
DC 
I- 10 















/ 


1 


11 














1 




1 j 


/ 














^1 


Q:/ 
J/ 

(f 


Or/ <i- 
oV 07 


/ 










f " 


j 


7 ' 

1 ) 


7 










/c 






\ 


// 












' ; 


> 


/ 

J 


f/ 


/ 










} 




/ 


A 


V 












f 


/ 


V 


'// 


/ 










^ 


y 


^A 


^ 


/ 












^ 


t^ 


^ 


X 














^ 


i>^ 







































20 40 60 

MILES PER HOUR. 

Fig. 7. 



80 



100 



ning at 60 miles per hour on a straight level track, assum- 
ing the weight of each car to be 50 tons and the cross- 
sectional area as 120 square feet. The tractive effort for a 



single-car train 



50 60 120(60)^ 
'^ V50 "^ 25 400 X 50 



= 31.1 pounds 



per ton; and for a three-car train is 



TRACTIVE EFFORT REQUIRED FOR CAR PROPULSION. 1 9 



50 



120(60)^ 



60 
Vsox"^ 25 400 X 50 X 3 



(-^") = 



I5.I 



pounds per ton. 

The formula for i?i is not suitable for car or train speeds 
less than about 5 miles per hour, for tests at speeds lower 
than this value have shown that the train resistance per 
ton is greater than the formula indicates. For example, 
tests by D. D. Ewing on a 27-ton interurban car showed 
that under varying track conditions a tractive effort of 
from 25 to 54 pounds per ton were required to start the 
car on a straight level roadway, the average being 40. To 
keep this car just perceptibly moving required 22 pounds 
per ton, which is about double the tractive effort needed 
at 5 miles per hour. 

The train resistance in tunnel and subway operation 
exceeds the value for trains run in the open because of 
increased windage. Various tests indicate that if the 
values derived from the preceding equation be increased 
by about 20 per cent., they will be found suitable for sub- 
way operation. 

9. Grades. — If grades be encountered additional trac- 
tive effort must be exerted. If a car be on a grade of 
inclination a to the hori- 
zontal plane, Fig. 8, the com- 
ponent of its weight along 
the direction of motion is 
W sin a, the other compon- 
ent being balanced by the 
reaction of the rails. To 
maintain uniform motion 
up the grade a force equal 
and opposite to W sin a must be exerted. For small values 




20 TRACTIOxN AND TRANSMISSION. 

of a, such as are met with in railway work, sin a — tan a 
approximately, whence grades may be expressed as the ratio 
of the vertical rise to the horizontal length of grade. It is 
customary, therefore, to consider that a grade of g per cent 
means a rise of g feet in a hundred feet. The tractive effort 
necessary to propel each ton of car weight up a one per 

cent grade is therefore X 2000, or 20 pounds, and to 

100 ^ 

draw a car of W tons up a grade of g per cent with uniform 

speed requires 

G = 20 qW pounds 

tractive effort. For a down grade G is considered negative. 
For routes having numerous grades, g should be taken 
as the equivalent percentage grade. On the assumption 
that only half of the kinetic energy acquired by a car or 
train in descending a grade is utilized in ascending the next 
up-grade because of stops on grades, non-alternate distri- 
bution of up- and down-grades, and the necessity of braking 
to avoid excessive speeds, the equivalent up-grade may be 
expressed as 

where }i\ and ^2 are the respective sums of all the rises on 
up-grades and the drops on down-grades in feet, and L is 
the length of the route in feet. In building a roadway, 
grades should be made as small as is economical, so that 
the cost of operation over the grades will be less than the 
interest on the amount necessary to reduce the grades. 

10. Curves. — Curvature of track presents additional 
resistance to the motion of a car because of increased 
flange friction. To neutralize this effect a larger tractive 
effort must be exerted, but since curves are usually of 
short length, this does not present a serious factor. Indeed 



TRACTIVE EFFORT REQUIRED FOR CAR PROPULSION. 21 

track curvature may be ignored in calculations of required 

torque unless such curves are numerous and very sharp. 

Sharp curves, such as occur with city traction systems, 

are generally rated by radius, but long curves are expressed 

in degrees, a one-degree curve being conventionally defined 

as one in which a chord loo feet long will subtend an 

angle of one degree at the center. Thus the radius of a 

J • •. ^1 s6o X loo - ^ 

one-degree curve is quite accurately ? or 5730 feet, 

2 T 

and consequently the number of degrees of curvature, c, 
of a curve, specified according to con- 
vention by radius i?. Fig. 9, is 

c = ^^ degrees. 

Curve resistance is usually taken as 

from 0.4 to i.o pound per ton of train 

weight per degree of curvature, and 

Fig. 9. depends upon the speed of the train. 

Prof. E. C. Schmidt gives the following formula for curve 

resistance as a result of tests on a 28-ton car: 

C = 0.058 cV pounds per ton, 

where V is the velocity in miles per hour. 

When a car moves around a curve it experiences a cen- 
trifugal force which depends in magnitude upon the speed 
and mass of the car, and the degree of curvature. This 
force tends to derail the car by rotating its center of mass 
outwardly around the outer rail. To neutralize this ten- 
dency the outer rail is raised above the inner rail to such an 
extent that the plane of the track is perpendicular to the 
resultant of the centrifugal and gravitational forces acting 
on the car. 
3 




22 



TRACTION AND TRANSMISSION. 



' Let m = mass of car in pounds, v = speed in feet per 
second, g = acceleration of gravity in ft./sec.^, and R = 
radius of curve in feet. Then 



R 



= horizontal centrifugal force, and 



mg = vertical gravitational force. 

An inspection of Fig. lo shows that the resultant of these 
forces will be perpendicular to the plane of the track when 
that plane makes an angle d 
with the horizontal such that 



e = tan~i 



Kg 




A road section devoid of 
curves is said to have a tan- 
gent track. 

1 1 . Acceleration. — In the 

foregoing paragraphs only ^^^' ^°' 

the torque to be exerted at the rim of the car wheels for 
uniform speed was determined. But in railway operation 
a number of stops must be made to allow passengers to 
board or alight from the cars, or to take on or unload 
freight, and further, between these stops the velocity of 
the car must be such as to maintain the specified schedule. 
Thus the car must be accelerated, and later brought to rest. 
To accelerate a car requires considerable tractive effort. 
The force in pounds acting on a body weighing w pounds 
which produces a change of velocity of a feet per second 



wa 



in one second is / = ^ = 



w 

2,2.2 



a pounds. Representing 



the weight of the car in tons by W, and the rate of accelera- 
tion in miles per hour per second by A, then the tractive 
effort required for acceleration alone is 



TRACTIVE EFFORT REQUIRED FOR CAR PROPULSION. 23 

^ 2000 W 5280 A 

F = • 7 7- = 91.3 WA pounds. 

32.2 60 X 60 

To allow for the energy of rotation of armatures, wheels, 
etc., which is difficult of exact determination and which 
depends upon the construction of these parts, the constant 
91.3 is replaced by 100. Acceleration rates of from | mile 
to 2 miles per hour per second are usual for cars and rapid- 
transit trains, while lower rates apply to locomotive-hauled 
trains. The greater the acceleration rate of a given equip- 
ment, the higher will be the schedule speed which can be 
maintained thereby. Limitations are imposed upon the 
maximum acceleration rate attainable by considerations of 
comfort to passengers, permissible starting current, and 
slipping of wheels on the rails. Thus the total tractive 
effort required at any Instant for the propulsion of a car 
or train of total weight W tons may be expressed by the 
complete general equation 

T = 50VIF + + -^^^ \ + 20qW 

I 25 400 L 10 J ^ 

+ 0.058 F TFc [ + 100 WA pounds. 

Representing the expression in braces, which includes the 
effects of train resistance, grades, and curves, by Tj, pounds, 
and rearranging, the acceleration becomes 

100 W 

12. Braking. — The kinetic energy represented by a 
moving car at any instant must be dissipated in some 
manner if the car is to be brought to a standstill at some 
later time. A force must in some manner be exerted 
between the roadway and the car, and must be in such a 
direction as to oppose and retard the latter 's motion. The 
force generally utilized is that due to static friction between 



r 



24 TRACTION AND TRANSMISSION. 

the wheel rims and the track rails where they are in con- 
tact. Two bodies with surfaces held in contact with 
each other by transverse pressure are capable of exerting 
forces upon each other along the direction of their plane 
of separation, which forces may be varied in magnitude 
from zero to such a maximum as will initiate slid- 
ing of the surfaces with respect to each other. This 
maximum usually bears a fairly constant ratio to the trans- 
verse force which presses the surfaces together, and is the 
coefficient of friction for the given materials of which the 
bodies are constituted. This coefficient for moving steel 
wheel rims on steel rails is, however, not constant because 
of the small areas in contact and the consequent enormous 
normal pressures, and because fresh surfaces are continu- 
ally becoming effective. This variable coefficient is also 
called the coefficient of adhesion ^ and, while it may amount to 
0.3 for clean dry rails, frequently sinks to 0.15 for wet rails, 
and may be subsequently raised to 0.25 by the application 
of sand. If the maximum retardation^ or negative acceler- 
ation, which this coefficient 0.25 will permit, be represented 
by Ab^ then the maximum retarding force or braking effort 

W 

Fb = 0.25 W = — Ab tons, 

g 
and consequently the retardation rate 

Ab = 0.25 g = 8.04 — '— = 5.5 miles per hour per second. 

To bring this frictional force into existence the kinetic 
energy of the car must be gradually dissipated. This is 
usually accomplished by pressing brake shoes upon the 
rims of the wheels so that the energy is consumed in attri- 
tion and heating of the shoes. The pressure on the brake 



TRACTIVE EFFORT REQUIRED FOR CAR PROPULSION. 25 

shoes is attained through levers actuated by hand, by 
pneumatic pressure, or by electromagnetic forces. The 
energy is sometimes allowed to expend itself in rotating 
the motor shaft against an electromagnetic counter-torque, 
a portion of the energy being thus returned to the line. 

The coefficient of friction between brake shoes and wheel 
rims decreases with increase of speed, of pressure, and of 
duration of application. The last is doubtless occasioned 
by the local elevation of temperature. To use the brake- 
shoe friction most effectually the pressure should, there- 
fore, be a maximum at high speed and be reduced with 
decreasing speed. This friction should never be so great 
as to cause slipping of wheels on the track, for the adhesion 
is thereby reduced and flat wheels may also result. 

PROBLEMS. 

3. Calculate the total train resistance of a New York Central locomotive 
weighing 220,000 pounds when it runs alone at a uniform velocity of a mile 
per minute. Cross section of locomotive is 120 square feet. 

4. Determine the tractive effort required to enable a train consisting of 
5 motor cars and 3 trailers to climb a 3.1 % grade with a uniform speed of 
15 miles per hour. The weight of the trucks per car is 9 tons; the weight 
of motors and control equipment per motor car is yj^ tons; and the weight 
of a car body is 21 tons. Each car can accommodate 80 passengers (aver- 
age weight = 140 pounds). 

5. If a curve having a radius of 1500 feet existed on this section of 
the road, how much additional tractive effort must be exerted to maintain 
the same velocity? 

6. Calculate the total tractive effort required to accelerate a car weigh- 
ing 30 tons, carrying 50 passengers, at the rate of 1.3 miles per hour per 
second on a tangent level track. Take 140 pounds as the average weight 
of a passenger. Neglect train resistance. 

7. Assume a train to be running on a straight level track at 60 miles per 
hour and an adhesion of 0.25 to be available for making an emergency stop. 
Find the elapsed time and distance covered in making the stop. 

8. Determine the proper elevation of the outer rail of a track for train 
speeds of 25 miles per hour, a curvature of 6 degrees, and a track gauge of 
4 ft. 83^ inches. 



26 TRACTION AND TRANSMISSION. 



J 



CHAPTER III. 

TYPES AND PERFORMANCE CURVES OF MOTORS. 

13. Traction Motors. — An electric motor suitable for 
traction purposes must exert the necessary torque for 
accelerating the car at the predetermined rate, or to pro- 
pel the car up a grade, without causing excessive energy 
>a) demands from the central station. This is possible only 

when large tractive efforts are exerted at low speeds, which 
follows from the fact that the power output of a motor is 
equal to the product of torque and speed. Torque depends 
upon the field flux and the current in the armature of the 
motor. The former varies with the field current, and, in 
an unsaturated motor, would be directly proportional to 
that current, but in practice it is less because of magnetic 
saturation. The speed of any motor depends upon the 
field flux, number of armature conductors, number of pairs 
of poles, and the counter electromotive force generated in 
the armature; thus 

{E - laR) 60 X 108 
^'^ ^ 2^55 ''^^' P^' ""'"•' 

where E is the impressed E.M.F., la is the armature 
current in amperes, R is the motor resistance in ohms, in- 
cluding armature and all series coils, p is the pairs of field 
poles, ^ is the flux per pole in maxwells, and 6* is the number 
of armature conductors in series between brushes. 



TYPES AND PERFORMANCE CURVES OF MOTORS. 2^ 

14. Direct-current Motors. — In a series direct-current 
motor the armature and field windings are connected in 
series and are traversed by the same current ; therefore the 
torque exerted is roughly proportional to the square of 
that current. If a small current flows, the field strength 
will be low, and from the foregoing expression for speed it 
is seen that the speed will be high. Again, if the motor 
takes a large current, the field strength will be intense 
and consequently the speed will be low. Thus, a series 
motor exerting large torque runs at low speed, and when 
exerting little torque operates at high speed. It follows 
that the power consumption of a series motor does not 
fluctuate violently, and therefore is well suited for rail- 
way work. 

In the shunt direct-current motor the field strength is 
approximately constant, and therefore the torque is directly 
proportional to the current and the speed is practically 
constant. When a large torque is required from such a 
motor its power consumption is enormous, since the speed 
is not materially lowered. Consequently the central station 
supplying equipment of this kind would be subject to great 
load variations. For this reason shunt motors are not used 
on railways. 

The direct-current series motor operating at 500 or 600 
volts has been in use since the advent of the electric railway. 
The commutating-pole series motor is now the standard 
for direct-current railways. Their use improves com- 
mutation, and permits the use of field control and high 
voltages. Fig. 1 1 shows the circuital relations of this motor. 

The tendency being to reduce the initial investment of a 
railway system, its operation, particularly over long dis- 
tances, must be effected at high voltages, since the principal 



28 



TRACTION AND TRANSMISSION. 



item of expense is the distributing system itself. But com- 
mutation difficulties limit the voltage of direct-current 
railway motors. Therefore it is usual to generate a high 
alternating electromotive force, preferably three-phase, at 
the power house, and to supply alternating current at this 
high voltage to a number of substations where, by means of 
transformers and converters, this current is changed to 



'TROLLEY p. A- 




Fig. II. 



direct current, which is then supplied to the railway motors 
over the low-tension distribution system. Such genera- 
tion and transformation entail large initial investment and 
operating expenses, and also considerable energy loss. 
These items may be greatly reduced by employing alter- 
nating-current motors, which can be operated at a potential 
of several thousand volts. 

15. Alternating-current Motors. — The advantages in- 
cident to the use of the alternating-current motor are 
the lower first cost of the low-tension distribution system, 



TYPES AND PERFORMANCE CURVES OF MOTORS. 29 

the substitution of the simple and efficient transformer 
substation for the converter substation, and the reduction 
of the cost of operation. It is not advisable to employ 
high trolley potentials in cities or densely populated sub- 
urban districts, but for trunk Hne operation, requiring 
an infrequent service, economical operation dictates high 
trolley potentials; in many cases transformation to a lower 
motor voltage is effected by transformers on the cars or 
locomotives. In alternating-current traction, controller sys- 
tems may be utilized which do not entail the large energy 
losses incident to starting direct-current motors. 

Three-phase generation is more economical than single- 
phase generation of E,M.F, The current from the former 
system may be converted into a two-phase current by means 




TROLLEYS 

Fig. 12. 

of a Scott transformer, each phase of which supplies single- 
phase current to the motors on one side of the station. 
Fig. 12 shows the scheme of connections. A single-phase 
load drawn from one phase of a large polyphase system 
unbalances the system for use otherwise. Phase converters 
or modifiers are used to restore this balance. 

There are several types of alternating-current single- 
phase railway motors at present in operation, but of these 
the compensated series motor is the only one used in this 
country. 



30 



TRACTION AND TRANSMISSION. 



Series Motors, — Consider a direct-current armature 
mounted within a single-phase alternating magnetic field, 
as in Fig. 13. When the armature is stationary an electro- 
motive force will be induced in the armature turns, due 
to the alternating flux which passes between the field 
poles. The greatest E.M.F.^s will be induced in the turns 
perpendicular to the field axis, since these turns link with 




Fig. 13. 



the greatest number of Hues of force; and no E.M.FJs will 
be induced in the turns in line with the field axis. The 
directions of the E.M.F.'s induced in the armature turns 
by the change in field flux are indicated in the figure by 
the full arrows, and it is seen that the maximum value of 
this E.M.F. is across BC, As in transformers, the effec- 
tive value of this electromotive force is 

JIt = — 7= ' 

V 2 10^ 

where $^ is the maximum value of the flux entering the 



(i) 



TYPES AND PERFORMANCE CURVES OF MOTORS. 3 1 

armature and A^ is the equivalent number of armature 

turns. 

The maximum number of Hues of force linked with a 

single turn depends upon the position of this turn in the 

magnetic field, and is proportional to the greatest value of 

$^ times the cosine of the angle of displacement of the 

turn from the position AD, Assuming the turns to be 

evenly distributed over the periphery of the armature, the 

average value of the maximum flux linked with the arma- 

2 
ture turns will be - <l>^. If there be Na conductors on the 

TT 

armature, the number of turns connected in continuous 

N 
series will be — ^- The electromotive forces induced in 
2 

the upper and lower groups of armature turns are added 

in parallel, consequently the effective number of turns in 

I Na Na 

series between brushes is - — - — — - . Therefore the 

22 4 

equivalent number of armature turns may be expressed as 

TT 4 2 TT 

Substituting this value of N in equation (i), the E.M.F. 
induced in the armature winding by the change in value of 
the field flux is 

Er = ■^' (3) 

and it lags 90"^ behind the field flux in time. 

If the brushes of the motor, A and D, are placed at the 
points shown in Fig. 13, this electromotive force will not 
manifest itself externally, since it consists of two equal 
and opposite components directed toward these brushes. 
This E.M,F, appears, however, in the coils short-circuited 



32 TRACTION AND TRANSMISSION. 

by the brushes, as will be shown later. The current, 
which enters the armature by way of the brush and which 
traverses the two halves of its windings in parallel, pro- 
duces an armature flux of maximum value ^am- This sets 
up a reactance E.M.F. in the armature which in the case 
of uniform gap reluctance can be similarly expressed as 

£a=%^' (4) 

V 2 10^ 

and lags 90° behind the current. 

When the armature revolves, there are, in addition, 
electromotive forces induced in the armature conductors as 
a result of their cutting the field flux. The directions of 
these E.M.F.^s are indicated by the dotted arrows, and it 
is seen that these E.M.F, ^s, generated by the rotation of 
the armature, add to each other and appear on the com- 
mutator as a maximum across AD. 

The average value of the electromotive force due to the 
rotation of the armature in a bipolar field is 

V 

Erotav = ^fNa — lO \ 
00 

where V is the armature speed in rev. per min. and <!>/ is 
the field flux; and the effective value of this E.M.F. is 

and is in time phase with the field flux, but appears as a 
counter E.M.F. at the brushes AD. 

When an alternating current is passed through the field 
coils, the alternating field flux is set up, and this flux pro- 
duces a reactive E.M.F. in the field winding lagging 90° 
behind the flux in phase, exactly as in a choke coil. The 
magnitude of this E.M.F. is 



TYPES AND PERFORMANCE CURVES OF MOTORS. 33 



V 2 10^ 



(6) 



where ^fm is the maximum value of the field flux, and Nj 
is the number of field turns. 

The electromotive force, £, which is impressed upon the 
motor terminals, is equal and opposite to the vectorial 
sum of Ea, Erotj Ef, and the IR drop of the armature and 
field windings, as shown in Fig. 14, where / is the current 




Fig. 14. 

flowing through the field and armature, and $ represents 
the phase of the flux. In this diagram, eddy current and 
hysteresis losses are ignored. The impressed electromotive 
force is therefore 

E = y/{Erot + IRy + {E^ + E,)\ (7) 

In the series motor, the same current passes through 
field and armature windings, and, if uniform reluctance 
around the air gap be assumed, then the armature and field 
fluxes will be proportional to the equivalent armature turns 
and field turns respectively. Representing by r the ratio 



34 TRACTION AND TRANSMISSION, 

of field turns to effective armature turns 

5am ^ iV_ ^ Nal2ir _ I 

whence ^/m = r^am- 

Substituting this value in (5), together with the equivalent 
of Ea/f to be derived from (4), there results 

-T.77 L 
Erot - j^«6o 

and Erot = 7- £/— • 

fr 60 

Neglecting the armature and field resistance drop, the 
impressed E.M.F. shown in (7) reduces to 



^ = ^V(g^/ + (^^ + ^)^ (9) 

which is the fundamental E.M.F. equation of the plain 
series motor. 
The power factor of the motor is 

COS = — = j===y (10) 

«°/V (£7)" + ('= + .)' 

and the current supplied to the motor is equal to the arma- 
ture voltage Ea divided by its reactance, still neglecting 
motor resistance. Solving (9) for Ea there results 

, Ea E 



^•V(^)'+ ("+')*' 



TYPES AND PERFORxMANCE CURVES OF MOTORS. 35 

When V = 60/, the motor is said to run at synchronous 
speed (bipolar field). The power factor of a plain series 



I 



motor, having r = i, when running at this speed, is - 

or 0.446, and for values of r other than unity the power 
factor is less than 0.446. It is true that if the resistance 
of the motor be considered, the power factor will exceed 
this value, but nevertheless it remains extremely low. 

The current intake under these same conditions is — =- 



When the motor is at standstill, V = o, and the power 

factor is zero. The current intake at standstill is — — -• 

2 Xa 
Hence the ratio of the current at synchronism to the cur- 
rent at standstill is — = -r- - = 0.894. The ratio of the 

V5 2 

torque at synchronous speed to the torque at standstill, 
since it varies as the square of the current, is (~p) -^ 

f-j = 0.80, which shows that the starting torque is but 

Kttle greater than the torque at synchronous speed. Since 
for railway service motors are required having large start- 
ing torque and which torque rapidly decreases as the speed 
of the motor increases, it is seen that independent of its 
low power factor, the plain series motor, having uniform 
magnetic reluctance around the air gap, is unsuitable for 
traction and for similar purposes. 

If, however, the reluctance of the air gap in the direction 
AD, Fig. 13, be increased, the power factor and speed- 
torque characteristics will be improved, and these will 
depend largely upon the ratio of field turns to effective 
armature turns, as will be seen by considering the construe- 



36 TRACTION AND TRANSMISSION. 

tion of the motor to be such that the proportion, equation 
(8), must be modified by introducing into its antecedents 
a constant considerably greater than unity. A motor 
of this kind, with few field turns compared to arma- 
ture turns, might be suitable for traction, but more 
important improvements have been made, which will now 
be discussed. 

It appears from Fig. 14 that the power factor of series 
motors may be increased by increasing IR and Erot^ or by 
decreasing Ef and £«. It is obvious that increasing IR 
signifies an increase in losses, thus resulting in a lower 
efficiency. Erot can be increased by increasing the number 
of armature turns. Both Ef and Ea can be decreased by 
lowering the frequency without affecting Erot^ hence low 
frequencies are desirable. To decrease the reactive elec- 
tromotive force of the field, it is necessary that the reluc- 
tance of the magnetic circuit be low, i.e., small air gap and 
low flux densities in the iron, in order that the required 
flux can be produced by a minimum number of ampere- 
turns. The armature reactive E.M.F., Ea, is not essential 
to the operation of the motor, and can be neutralized by 
the use of compensating windings, and this feature of 
alternating-current series motors is a very important one. 

The compensating winding is embedded in slots in the 
pole faces, as shown in Fig. 15, which represents a West- 
inghouse four-pole compensated single-phase railway motor 
with its armature and field windings removed. The num- 
ber of turns of the compensating winding is adjusted so 
as to set up a magnetomotive force equal and opposite to 
that due to the current in the armature coils. The com- 
pensating winding may be energized either by the main 
current, by placing this winding in series with field and 



TYPES AND PERFORMANCE CURVES OF MOTORS. 37 



armature, or by an induced current, which is obtained 
by short-circuiting the compensating winding upon itself, 
thus utihzing the principle of the transformer in that the 
main and induced currents are opposite in phase. The 




Fig. 15. 
former method of neutraHzing £« is known as conductive 
or forced compensation, and may be used with both alter- 
nating and direct currents, and the latter method is known 





A(\A/ 






Fig. 16. 



Fig. 17. 



as inductive compensation^ and may be used only with alter- 
nating current. 

Figs. 16 and 17 show schematically the connections of 
the conductively and inductively compensated alternating- 
current series motors respectively. The compensating 
winding is preferably distributed so that the armature 



38 TRACTION AND TRANSMISSION. 

reactance is neutralized as completely as possible. The 
current flows in the same direction in all of the conductors 
of the compensating winding embedded in one field pole, 
and flows in the opposite direction in the conductors em- 
bedded in the adjacent poles. 

When the compensating winding completely neutralizes 
the armature reactance, the impressed electromotive force 
from equation (7) is 

E = ^(Erot + IRY + Ef\ (12) 

where R is the resistance of the motor including that of the 
compensating winding. If R be neglected, then, since 



£^- = ^-7:^/ and E = Eryj(^~ J+ i, 



60 /r 



the power factor is 



COS (/> = —~ = , (13) 

E VF2 + (6o/r)2 ^ ^^ 



The motor current is 



/ = — . ^ (14) 



-VL-^) 



K60 frJ "^ ^ 

At synchronous speed F = 60/, and therefore the power 
factor at this speed becomes , 

Vl + t2 

Still neglecting the motor resistance, the current intake 

Er 

at synchronous speed is — — ^ and at standstill it 

is — y consequently the ratio of the current at synchronous 



TYPES AND PERFORMANCE CURVES OF MOTORS. 45 



























/ 


2000 


A n 






















/ 






















A 






9 ^ 




















A 
















% 


Fpr 






/ 






1500 


3.0 




y 










[£it^( 




/ 












\ 








f 


f/ 






^-. 





-5o 


/ 




\ 








f\ 


f 








u 

UJ 

> 


X 

a: 






\ 






^J 


/ 










§1000 


a. 

CD 

-2-0 "^ 








\ 
















i 








\ 


N 














OQ 












/ 


\ 


^ 




















/ 


/ 




s> 


^ 








500 


lO 








/ 






















A 




















R 




/ 


/ 






50 

6 


H.R 
500 \ 


MO 

OLTS 


FOR 

3, 












/ 








3 

GEA 


3"W 
={ RA 


HEEL 

no 1 


s. 

^-69 












/ 





















100 



90, 






80' 



70 



20 



40 



60 
AMPERES. 

Fig. 23. 



80 



100 



120 



46 



TRACTION AND TRANSMISSION. 





























/ 


























/ 


/ 


























/ 


















% E 


-FICIENCY 




/ 


/ 






-7000- 


/ 


^ 


^ 














/ 




, 90j 


-48 


I 


/ 
















/ 








6000T" 
















/ 


/ 






GO 






\ 












,o^ 


/ 










40 


5000" 


\ 












/ 










70 

1- 

z 


=) 
-32-0- 

III 




k 










-V 


/ 










o 
d: 




Li_ 
Ld 

U 


\ 








/ 












CL 


-24-^ 

2 


-3000- 


> 

r ^ 


\ 


\ 




/ 
















< 

DC 




N 


>1 


/ 




















DQ 

_J 




/ 


/ 






SP 


^ED__ 






















/ 


/ 


























/ 










200 
5 


H.P. 
50 V 


MC 
OLTS 


TOR 









1000- 


-^ 












3 
GEAF 


3"Wf 
\ RA" 


^EEL 
•|0 2 


3, 
0-63 



































75 



150 



225 3C0 

AMPERES 

Fig. 24. 



375 



450 



525 



TYPES AND PERFORMANCE CURVES OF MOTORS. 47 









\ 






















-80— 








i 


























\ 


















/ 


-70 — 








A 




■^ 


^ 


^^c 


tnr. 






/ 


/ 










V- 




£_ 


£FFI, 


;v^ 


-^ 




/ 




-60 






/ 


/ 


\ 








y 


y 


'^ 


■\ 




H 






\ 


t-. 








/ 


^ 




,..80 


DC 

=> 
-50-O- 


-2500- 




Li_ 








Y 


V 






/ 






LiJ 


> 










\ 


V 


/ 








u 

CL 


Q. 
CO 

-40-y- 

2 


-2000- 


1- 

< 












> 


/ 










co 












/ 


^V 


^^ 






60 






CD 

-I 










f 












-^ 














^y 


























f 


















20 








/ 


/ 


























/ 








250 


H.P 
25 C> 


. MC 

'CLES 


)TOR 












/ 










225 \ 
6|2"W} 


'OLTS, 
-lEELS. 



































250 



500 



750 1000 

AMPERES 

Fig. 25. 



1.250 



1500 



1750 



48 



TRACTION AND TRANSMISSION. 





























/ 


-240 


•^ 






















/ 
























/ 








■\ 


/ 


/^ 


'jEFl 


ICiE-t 


slC"/ 








■~A 


/ 




rsfs 




.-< 


F^^ 












/ 


/ 


\^ 


^v^^^ 








/ 


/ 








-1-8 0( 


> 


/ 


/' 










/ 


/ 








Qn 




/ 












/ 










ou 






/ 










/ 














-150C 


DQ 

_J 


/ 








J 

,w- 


/ 












f-7-0 
z 
u 
f "> 


-1-2-0 C 


UJ 


( 








<s/ 
















o 








/ 






25 


H 

300 


p. I 

vo 


LTS 


3R 


^dl- 


-90(: 








/ 


/ 








2 


5 CY 
3 PH 


CLES 
ASE 






) 






/ 


















u 








/ 






















bOC 


1 


/ 


/ 






















-30( 


J 


/ 










^ 


^' 


.^^ 








o 


/ 


/ 




^ 




.-^'^ 


















/ 



























10 20 30 40 50 

AMPERES PER PHASE 

Fig. 26. 



60 70 



TYPES AND PERFORMANCE CURV^ES OF MOTORS. 49 

2 TvegT' = 2 7rDnpT/24.ng foot-pounds. 
T = 2^ngegT^/npD pounds. 

Equating the effective power exerted by the motor to the 
power exerted by the tractive effort, 

2 7rVmegT'/2S,ooo = 5280 Fr/6o- 33,000 horsepower. 

V == 0.0714 — — Vm miles per hour. 

Figs. 23 to 26 show respectively the characteristics of 
G.E. 216A motor, of the motor used by the I.R.T.Co. of 
N.Y.C., of the Westinghouse compensated single-phase 
motors used by the N.Y., N.H. & H.R.R., and of an 
induction motor. 

PROBLEMS. 

9. Plot a curve showing the ratio of the current taken by a compensated 
series motor at synchronous speed to that taken at standstill, coordinated to 
the ratio of the number of field turns to the effective armature turns. 

10. The motor of an electric car having 33-inch wheels, when traveling 
at 25 miles per hour, exerts a torque of 550 pounds at one foot radius from 
the center of the armature shaft. If the gear ratio be 26 to 60, and the effi- 
ciency of the gears be 97 %, determine the tractive effort at the base of the 
car wheels, the horsepower, and the number of revolutions of the motor 
per minute. 

11. Determine the horsepower output and speed of the induction motor 
wdiose characteristic curves are given in Fig. 26, when taking 50 amperes 
at 2850 volts. How many stator poles has the motor? 

12. The gearless 25-cycle, single-phase motors used on the New Haven 
locomotives have 12 poles. Determine the velocity of the locomotives, 
which have drivers 62 inches in diameter, when the motors run at synchron- 
ous speed. 

13. The total weight of a Pennsylvania electric locomotive is 166 tons, 
of which 104 tons are carried by the drivers, and the trailing load is 550 
tons. What is the maximum grade this train can ascend with uniform 
velocity without slipping the wheels on clean dry rails? Neglect train 
resistance. 



50 TRACTION AND TRANSMISSION. 



CHAPTER IV. 

SPEED CURVES. 

i8. Motor Limitations. — The size of the motors to be 
installed on cars so that they may perform a proposed 
service must be such that the motors will exert the necessary 
tractive effort for the prescribed acceleration and operate 
without overheating. As the tractive effort exerted by a 
motor depends upon its current intake, and the maximum 
current which may be supplied to the motor depends upon 
commutation, it is seen that the rate at which a car may 
be accelerated is dependent upon the allowable current 
input. Another limitation to the rate of acceleration, 
besides the consideration of comfort to passengers, is ex- 
pressed by the coefficient of friction or adhesion, that is, 
the ratio of the tractive effort necessary to cause slipping 
of the wheels on the rails to the total weight on the drivers. 
This coefficient depends upon the condition of the track. 
The following values are approximate and are based upon 
a uniform torque exertion: 

Clean dry rails 0.30 

Wet rails 0.18 (with sand 0.25) 

Sleet-covered rails 0.15 (with sand 0.20) 

Snow-covered rails ' o. 10 (with sand o. 15) 

It is seldom necessary to apply motors to every axle, 
economy dictating that the number of axles equipped be 
as small as possible and as permitted by the coefficient of 
adhesion. In train operation some cars are equipped with 
motors while others are mere trailers without motors. 



SPEED CURVES. Si 

The heating of motors in service is determined by the 
square root of the mean square current suppHed to the 
motor and the average voltage across the motor terminals. 
This mean square current is obtained from a series of in- 
stantaneous current values taken over a considerable time 
interval, as shown later. Thus, a motor should be selected 
which will commutate the abnormal current taken during 
the period of acceleration without excessive sparking at 
the brushes and also perform the required service without 
excessive temperature rise. 

19, Motor Capacity. — To determine the motor capac- 
ity for a proposed service, a knowledge of the load under 
which the motor must operate is essential. This load is 
of an exceedingly variable nature, fluctuating between no 
load at stopping points and a maximum load, which occurs 
during starting of the car. The method of procedure is 
as follows: a trial equipment is assumed (a guide to its 
selection may be obtained from a comparison of the equip- 
ments of similar existing installations), and from the motor 
performance curves there are plotted curves of speed of 
the car in traversing the entire roadway and_of motor 
current. The former curve enables one to foretell if the 
prescribed schedule speed can be maintained, allowing a 
reasonable margin for making up delays, and the latter 
curve serves as the basis for determining whether the 
assumed motor can perform the required service without 
such extreme heating as to endanger the insulation. 

20. Speed. — The velocity of a car in operation varies 
widely from time to time. Starting from standstill, the 
car is accelerated, rapidly at first, then more and more 
slowly until a uniform speed is attained. After running 
at this speed for a definite time, the current is turned oU 



52 TRACTION AND TRANSMISSION. 

and the car is allowed to coast, the velocity meanwhile 
gradually decreasing. Finally the brakes are applied in 
order to bring the car rapidly to rest at the next stopping 



%/• i 




. vvw€, Fig. 27. 



point. Here freight or passengers are taken on or dis- 
charged; thereafter similar runs are performed. 

21. Typical Speed Curves. — The velocity of a car at 
successive instants of time may be graphically portrayed 
by a speed curve, in which the instantaneous speeds are 
plotted in terms of time. Such a curve takes the form 
of a series of lobes, each one representing a run and one of 
which is shown in Fig. 27. The slope of the curve at any 
point indicates the time rate of change of speed. This 
slope may be positive, zero, or negative, corresponding 
respectively to acceleration, uniform speed, or retardation. 

The speed curve may be considered as made up of four 
parts as follows: starting, motor, coasting, and braking. 
The starting part corresponds to the period of manipula- 
tion of the controller, the acceleration of the car and the 
current in the motor being kept constant, while the voltage 
impressed upon the motor is gradually increased from zero 
to its normal value. The motor part corresponds to a 



SPEED CURVES. 53 

gradual decrement of acceleration of the car and of motor 
current, normal voltage being impressed upon the motor. 
The coasting part corresponds to the movement of the car 
under its own momentum, no current passing through the 
motor. The braking part corresponds to the period during 
which the car is being quickly brought to rest by the 
absorption of energy at the brake shoes. The starting 
and motor parts are often considered together as constitut- 
ing the acceleration part of a speed curve. 

The ordinate B of the speed curve represents the max- 
imum velocity of the car during the particular run, and the 
horizontal line DE shows the duration of standstill at the 
subsequent stop. The schedule speed of the car is obtained 
by finding the area of the speed curve over the entire road- 
way and dividing by the total time taken therefor inclusive 
of stops. This time is the interval between A of the first 
run and E of the last one. The shorter the time of stops 
the greater will be the schedule speed, other conditions 
remaining unaltered. The greater the rates of acceleration 
and retardation the greater will be the schedule speed pro- 
vided the same maximum speed is attained. If the rate 
of braking be too high the car wheels will slide on the rails, 
and there will be a tendency for the car body to move ahead 
over the trucks. The maximum practicable braking rate 
is considered to be 2.5 miles per hour per second. 

22. Data for Plotting Speed Curves. — The plotting of 
a speed curve for a proposed equipment over a typical run 
requires a knowledge of the following conditions: 

Type of motor. 

Number of motors per car or train. 

Motor performance curves at full line voltage and at 

a definite gear ratio, 
5 



54 TRACTION AND TRANSMISSION. 

Total weight of the car with live load, 

Plan and profile of the roadbed, 

Schedule speed required, 

Rates of acceleration and braking, and 

Duration of stops. 
For single-car operation (double-truck cars) a four- 
motor equipment is preferable, whereas for train operation 
two-motor equipments are generally used, and sometimes 
both motors are placed on one truck. 

The performance curves of a railway motor show its 
characteristics at normal voltage under any load. When 
starting the series motor, the voltage impressed upon its 
terminals is low at first, and is gradually increased by means 
of a controller, which cuts out resistance or, with single- 
phase motors, decreases the ratio of transformation of a 
compensator. With suitably designed controllers properly 
operated the current supplied to the motors will be roughly 
uniform until the full line voltage is impressed upon the 
terminals of each motor. The torque exerted, being pro- 
portional to the current intake, will also be uniform. After 
the line voltage is applied to the motors, their performances 
are entirely dependent upon their characteristics. 

It is essential to have a reliable estimate of the weight of 
the tentative car for a proposed service, this weight to 
include live load, electrical equipment, and brake apparatus. 
Weights of car bodies and trucks are given in Chapter I. 
The average weight of passengers may be taken as 140 
pounds per individual. The weights in pounds of some 
standard 600-volt electrical equipments having commu- 
tating-pole railway motors, made by the General Electric 
Co. and the Westinghouse Electric & Manufacturing Co., 
for direct-current railways follow: 



SPEED CURVES. 



55 







Weight of Each 


No. of 


Type of 
Control. 


Weight of 


Total 


Trade Name. 


H. P. 


Motor Including 


Motors 


Control 


Weight of 






Gears and Case. 


xf X v/kvy& J* 




Apparatus. 


Equipment. 




258 


25 


885 


2 


K-63 


750 


2520 










4 


K-35 


1285 


4825 










4 


PCt 


1335 


4875 




247 ... . 


40 


1740 


2 


K-63 


755 


4235 


c3 








4 


K-35 


1490 


8450 


^u 








4 


PCt 


1490 


8450 


'C> 


2I6A* . . 


50 


2885 


2 


K-ii 


1015 


6785 


2 








4 


K-14 


2250 


13790 


s ■ 








4 


t 


2070 


I361O 


'S 


203 


50 


2280 


2 


K-36 


IIOO 


5660 










4 


K-35 


1520 


10640 










4 


PCt 


1600 


10720 


O 


201 


65 


2845 


2 


K-36 


1315 


7005 










4 


K-35 


1690 


13070 










4 


PCt 


1650 


13030 




240 


105 


3840 


4 


^■A^ 


2820 


18180 








4 


PCt 


1710 


17070 




' S06A . . . 


25 


913 


2 


K-63-B 


750 


2576 


c3 








4 


K-35-G-2 


1228 


4880 








4 


HLt 


1610 


5262 


5;^ 


514C... 


40 


1795 


2 


K-63-B 


822 


4412 


^ 








4 


K-35-G-2 


1550 


8730 










4 


HLt 


1610 


8790 . 


532B... 


50 


2340 


2 


K-36-J 


1156 


5836 


6 








4 


K-35-G-2 


1654 


IIOI4 


W 








4 


HLt 


1769 


1 1 129 


<u 


306CV4. 


65 


2675 


2 


x.^-3^>[ 


1191 


6541 


:3 









4 


K-35-G-2 


1894 


12594 


too 








4 


HLt 


2032 


12732 


.S 


548C . . . 


100 


3195 


2 


K-3^G-2 


1642 


8032 


CO 








2 


HLt 


1645 


8035 


^ 








4 


HLt 


2654 


15434 




557A8 . . 


140 


4050 


2 


HLt 


2105 


10205 



* Without commutating poles. t Multiple unit. 

The weights of single-phase motors somewhat exceed the 
foregoing values for the same capacity, but owing to their 
limited adoption up to the present time, the design of this 
type of motor has not yet become standardized. 

The dimensions of the car chosen for the proposed rail- 
way should be known, particularly those dimensions which 
limit the minimum permissible radius of track curvature, 



56 



TRACTION AND TRANSMISSION. 



the clearances on each side of the track at curves, and the 
maximum possible size of motor which can be installed on 
the truck. 

The physical characteristics of a roadway are usually 
embodied in a map and profile of the route showing the 
length of line, proposed regular stations, junctions and 
crossings with existing roads, switches and branch lines, 
and the location and extent of grades and curves. 

A subdivision of the total length of the road into city, 
suburban, and interurban sections can usually be effected. 
Different operating conditions obtain in these sections, 
because the schedule speeds and length and frequency of 
stops are not the same for all. Representative values for 
these factors follow. 



Service. 



Interurban express 

Interurban local 

City rapid-transit express . 

Suburban 

City elevated or subway 

(local) 

City surface lines 



Schedule speeds 

in miles per 

hour. 



35 to 60 
25 to 40 
20 to 30 
15 to 20 

15 to 20 
8 to 12 



Average dura- 
tion of stops 
in seconds. 



60 
30 
25 
15 

12 

7 



Number of 
stops per mile. 



0.05 to O. 2 
0.3 to O. 7 
0.4 to I.O 
I to 2.5 



to 3 
to 10 



The choice of gear ratio for the trial equipment should 
be such that^ the peripheral velocity of the motor armature 
when the car is running at its highest speed will not be 
excessive. The ratio of the maximum speed to the schedule 
speed varies between 1.2 and 1.8, this ratio increasing as 
the runs become shorter and the duration of stops becomes 
longer. This enables the selection of the proper gear ratio. 

23. Plotting Speed Curves. — To understand the method 
commonly used in plotting speed curves consider the dif- 



SPEED CURVES. 



57 



ferent portions of the curve in Fig. 28 and the following 
formula for the car acceleration (see § 11): 

Tm ~ Tt 



A = 



(I) 



100 W ' 

where Tm is the tractive effort exerted by a motor, Tt is the 
train resistance per motor, and W is the weight of the car 
or train per motor. Then 

Tm = Tt+ 100 WA. (2) 

The starting part of a speed curve is taken as a straight 
line, and it passes through O, the origin of time, at an 
angle Oa with the horizontal such that Sa = tan"M, where 




Fig. 28. 

A is the assumed constant rate of acceleration at starting. 
It terminates at the point A having a speed ordinate taken 
from the motor characteristic curves for full voltage cor- 
responding to the tractive effort T^ calculated from equa- 
tion (2), in which Tt is based on half schedule speed. 

The motor part of the speed curve is considered as made 
up of a series of elements which are. themselves straight. 
The speed ordinate of the upper end of any element is 



58 TRACTION AND TRANSMISSION. 

assumed, while that of its lower end is the same as for the 
upper end of the preceding element. This element makes 
with the horizontal an angle 6^ = tan~^^„, where An 
is the average of the accelerations corresponding to the 
speeds at the terminals of the element and each calculated 
by means of formula (i). The calculation of these ele- 
ments is greatly facilitated by two auxiliary curves, one 
showing the relation between motor tractive effort and speed 
and the other between train resistance and speed. 

The coasting part is generally assumed to be straight, 
although it really is concave towards the time axis. It is 
drawn from an assumed point B and makes with the hori- 
zontal an angle dc = tan~^ Ac, where Ac is calculated 
from formula (i), whose terms are based upon the speed V 
which is the ordinate of the point B. The other end, C, of 
this part of the curve is determined by the intersection with 
the remaining part. 

The braking part of the speed curve is also assumed to 
be straight, passes through the time axis at D corresponding 
to the specified expiration of the run, and makes with the 
horizontal an angle 63 = tan'M^, where AbIS the assumed 
rate of braking. 

The method just outlined assumes that equal distances 
along the two axes have the same numerical value; for 
example, if one inch along the horizontal axis of Fig. 28 
represents 50 seconds, then one inch along the vertical axis 
must correspond to 50 miles per hour. If, for convenience, 
other than equal value scales are used, the parts of the 
speed curve cannot be constructed by laying off the angles 
as described, but instead, by measuring the horizontal and 
vertical components on cross-section paper according to 
their individual scales. Thus, in laying out the motor part 
of a speed curve, the abscissa increment, in seconds, for an 



SPEED CURVES. 



59 



element may be determined by dividing the speed incre- 
ment in miles per hour by the average acceleration in miles 
per hour per second, or symbolically A/ = AF/^. 

24. Numerical Example. — The process of plotting a 
speed curve is best illustrated by considering a specific 
case, as follows: 

(a) Data. Car, single car to seat 40 passengers and to 
accommodate an equal number standing, weighing with 
trucks 23,650 pounds. Cross section, 5 = 95 square feet. 



900 
CO 800 



3 700 



Q- 

z 600 

§500 
ii. 

^400 

LJ 

> 

H300 

< 

1-200 

100 








\ 
























\ 


\ 
























\ 
























\ 


\ 
























\ 


























\ 


























\ 


s 
























\ 


\^ 


























^^ 


-~- 





























10 



20 30 

SPEED IN MILES PER HOUR. 

Fig. 29. 



40 



Trial equipment: four direct-current 50-horsepower, 
600-volt G.E. 216A motors with Type K-14 control; total 
weight is 13,790 pounds, § 22. Characteristic curves of 
motors are shown in Fig. 23 for a gear ratio of 17 to 69. 
From these curves a new curve, Fig. 29, of tractive effort 
per motor and speed is plotted for convenience. 



6o 



TRACTION AND TRANSMISSION 



Run, 0.8 mile run on a straight level track. Schedule 
speed = 20 miles per hour. Length of stop = 20 seconds. 
Initial acceleration rate = 1.5 miles per hour per second. 
Braking rate = 2 miles per hour per second. 

The total weight of the car with live load is 

W = 23,650 + 13,790 + (80 X 140) = 48,640 pounds 

= 24.32 tons. 
The total train resistance is 

i— WV SV^ 
^T = 5o^W + ~— + — = 246 + 0.97 V + 0.238 F2 . 

Fig. 30 shows the relation which exists between the train 
resistance per motor Tt = ry/4 pounds and the speed V 
miles per hour. 



160 

CD 
Q 
























^ 






















^ 


y 


Z 



^120 

z 

III 
















^^ 






















^ 












ESISTANC 

00 











-^ 


















-^^ 






















oc 

z 

< 40 

oc 














































































10 



20 30 

SPEED IN MILES PER HOUR. 



40 



Fig. 30. 



SPEED CURVES. 6l 

(b) Acceleration at Subnormal Voltages, To produce an 
acceleration of 1.5 miles per hour per second requires a net 
tractive effort of 

100 WA = 100 X 24.32 X 1.5 = 3648 pounds, 

or a net tractive effort of 3648 -^ 4 = 912 pounds per motor. 
To neutralize train resistance during the period of initial 
acceleration additional tractive effort must be exerted. 
The amount may be taken equal to the train resistance at 
half schedule speed, that is, at 10 miles per hour. Its value 
is found from Fig. 30 to be 70 pounds per motor. There- 
fore the total tractive effort to be exerted by each motor in 
starting is 

Tm = 912 + yo = 982 pounds. 

This tractive effort is produced when each motor takes 
64 amperes at 600 volts^ as shown by the motor performance 
curves, Fig. 23 ; and the corresponding speed of the car is 
16.9 miles per hour. Thus, the current consumed as the 
car is accelerated uniformly at the prescribed rate from 
standstill to a speed of 16.9 miles per hour is maintained 
roughly constant by the controller at a mean value of 
64 amperes. The time required to attain this speed is 

-7 = — — =11.3 seconds. This represents the first point 
A 1.5 

of the speed curve, and is shown at A in Fig. 31. Since the 

acceleration during the first 11.3 seconds of the run was 

approximately uniform, the speed curve over this interval 

may be drawn as a straight line, as OA . 

(c) Acceleration at Normal Voltage, The full line volt- 
age is applied to each motor when the speed of 16.9 miles 
per hour is reached, and thereafter the acceleration be- 



62 TRACTION AND TRANSMISSION. 

comes less and less because the current decreases as the 
car speeds up and this results in a lower available tractive 
effort. Increased train resistance at higher speeds is also 
instrumental in lowering the acceleration rate. To obtain 
other points of the speed curve, the car is supposed to be 
running at some higher speed, say 20 miles per hour. At 
this speed the motor current will be 48.2 amperes, the 
total tractive effort will be 660 pounds per motor, and the 
train resistance will be 90 pounds per motor. The net 
tractive effort producing acceleration is 660 — 90 = 570 
pounds; whence the rate of acceleration at a speed of 20 
miles per hour is 

A T^ — Tf I 24. ^2\ ., , 

A^= TTr/ =57Q-^ looX ~^^^ = 0.94 mile per hr. per sec. 
100 W \ 4 / 

The average acceleration during the period in which the 
velocity of the car increased from 16.9 to 20 miles per 
hour may be taken without serious error as the mean of 
the initial and final acceleration rates of the period. The 
time required to gain this velocity increment is, of course, 
the increment divided by the average acceleration, which 
in this case is 

20 — 16.9 3.1 J 

A/ = ~ = -^^— = 2.54 seconds. 

1.5 + 0.94 1.22 

2 

Thus, the second point of the speed curve shows a veloc- 
ity of 20 miles per hour at 11.3 + 2.54, or 13.84 seconds 
(6, Fig. 31). 

This process is continued with small velocity increments 
until the speed of the car becomes constant. A tabula- 
tion of the values so obtained follows; the various points are 
indicated on the curve. 



SPEED CURVES. 



(>i 



















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1 






















































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1 






















































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1 






















































1 






















































1 






















































1 




































LU 


















1 






















































1 






















































1 

1 






















































1 

f 






















































1 
1 




































Q 


















1 
1 






























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-^' 


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' 




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u. 


'^ 












1 
































































































, 












1 










































1 










/ 












































I 










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1 














































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1 
















































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1 
















































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1 


















































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1 




















































\ 






















































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u^ 


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i 






















































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\ 


Oi 






















































\ 






















































\ 


^ 






















































v 


» 




















































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^; 


e 




















































T 


^ 


J 

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K 


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'liJ (zt 
CO 



O 



o o 

CO CM 

anoH a3d S3nm 



64 



TRACTION AND TRANSMISSION. 



Point. 


Speed, 
V. 


Tractive 
effort. 


Train 

resistance, 

Tt. 


Net trac- 
tive effort, 


Accelera- 
tion rate, 
A. 


Total 
time. 


A 


16.9 

20 

22 

24 
26 
28 
30 
32 
35 
36.8 






912 
570 
434 
328 
252 
185 
133 
90 

25 



I-50 
0.94 
0.714 
0.540 

0.415 
0.304 
0.219 
0. 148 
0.041 



11.30 

13-84 
16.26 

19 -45 
23.65 
29.22 
36.88 
47 78 
79.6 
177.0 


b 
c 
d 
e 

f 
g 
h 

i 
J 


660 
530 
430 
360 
300 

255 
220 
170 
152 


90 

96 

102 

108 

115 
122 

130 
145 
152 



(d) Braking, After plotting the entire acceleration curve 
of a car with an assumed electrical equipment for a partic- 
ular run, the speed curve is completed by drawing the 
coasting and braking curves. Since the time of passage 
over a setction of the road is specified by the schedule 
speed and the average duration of a stop, it is necessary 
to construct the braking curve first so as to determine how 
much coasting may be permitted and still bring the car 
to the next station in the required time. 

In the numerical illustration the car is to travel 0.8 mile 
at a schedule speed of 20 miles per hour, which means that 

the time required for this run is =144 seconds. 

20 

But this time includes a stop of 20 seconds; therefore the 
actual running time is 124 seconds. The braking curve 
may now be drawn through this point on the time axis 
at a slope corresponding to the braking rate and extending 
to its intersection with the acceleration curve at F. It 
should be drawn as a straight line, and, since the braking 
rate is specified at 2 miles per hour per second, the line 
will pass through the point which indicates that the veloc- 



SPEED CURVES. 65 

ity of the car is 2 X 10 = 20 miles per hour at a time of 
124 — 10 = 114 seconds from the beginning of the run. 

(e) Coasting, Since the ordinates of a speed curve are 
velocities and the abscissae are times, the area of such a 
curve will be expressed in units of velocity X time, or 

— : X time, or simply in units of distance. Thus, 

time 

in Fig. 31, the area of a large square is 10 miles per hour 

X 20 seconds = 200 mile-seconds per hour = -3^0% or rV 

mile. The area enclosed by a speed curve is therefore a 

measure of the distance traversed by the car. 

The speed curve drawn thus far allows for no coasting, 
and the area enclosed thereby may be less than, but in 
general will exceed, that representing a run of 0.8 mile. 
For a run of this length the speed curve must enclose 
exactly 0.8 -^ tV = 14.4 large squares. In order to obtain 
just this area, the position of the coasting curve BC is 
varied until properly located; its slope, however, cannot 
be taken at random. 

When the current supply to the motors is discontinued 
the car tends to run at constant speed, but train resistance 
retards the motion and produces a negative acceleration. 
As train resistance depends upon the speed, the coasting 
curve will not be strictly a straight line, but will have a 
slight curvature tending to become more nearly horizontal 
at lower speeds. It is usual to draw the coasting line 
straight and at a slope corresponding to the train resistance 
value at the speed at which the car is running when the 
power is cut off. 

The coasting curve is drawn at the proper inclination 
in a trial position and the resulting area of the speed curve 
is determined. If the area be different from the proper 



66 TRACTION AND TRANSMISSION. 

value the line is shifted parallel to itself up or down as the 
case may be, until the enclosed area is found to be correct. 
Should the coasting curve require considerable shifting so 
that it commences at a somewhat different speed value, 
then its inclination must be redetermined on this basis. 
The area of the curve AFD of Fig. 31 is 16.8 large squares, 
and the position of the coasting curve was adjusted so 
that the enclosed area ABCD is equal to 14.4 squares 
thus the speed curve truly depicts a 0.8 mile run. The 
train resistance at the speed where coasting begins is 130 
pounds per motor. The negative acceleration produced 

thereby is 7^- r = 0.21 mile per hour per 

100 X (24.32 -T- 4) 

second, a value giving the proper slope of the coasting line. 

Had the area of AFD been less than 14.4 squares, the 
curve would have indicated that the chosen equipment is 
incapable of maintaining the specified schedule speed under 
the given conditions. In such cases other curves should 
be drawn for the same equipment with lower gear ratios, 
or for other equipments comprising larger motors. On 
the other hand, if the excess area be unduly large, other 
speed curves corresponding to higher gear ratios or smaller 
motors should be constructed. A reasonable margin 
should, however, be allowed for making up for delays. 
The equipment ultimately selected for the given service 
shouH be able under emergency conditions to make a 
complete trip in 5 to 15 % less running time than that 
allowed for regular service. 

25. Distance Curves. — Speed curves of cars over runs 
having grades or curves are more difficult to construct 
than those over a tangent level roadway. Here the addi- 
tional tractive effort required for propelling a car or train 



SPEED CURVES. 67 

up a grade or around a curve must be considered, and 
indeed, these additional forces are applied at definite 
places on the run. This implies a knowledge of the exact 
location of the car at every instant of time, so that these 
influences may be properly represented on the speed curve. 
The instantaneous positions of a car are shown most con- 
veniently by a distance curve plotted in terms of time. 

The distance curve for the run mentioned in the fore- 
going is plotted as follows: The average velocity over the 
first 1 1.3 seconds of the run is i (o -f 16.9) = 8.45 miles 
per hour, and therefore the space traversed during this 

.J. 11.3 X 8.4s ., 95.4X5280 ^^ . 

period IS — ^- ^ mile, or ^^ ^ ^ = 95.4 X 1.467 

3600 3600 

= 140 feet. The average velocity over the next 2.54 seconds 
is ^ (16.9 -f 20.0) = 18.45 niiles per hour, and the dis- 
tance traveled during this time interval is 18.45 X 2.54 X 
1.467 = 68.6 feet. This process is continued over the 
entire running time, and the final sum should be equal to 
0.8 X 5280 = 4224 feet. The speed and distance curves 
are generally plotted si- 
multaneously, using for 9 j ,850^ \ ^ 



convenience the same time r- soo— *i — ^ 

+ 2.3^ GRADE ^>| 

increment values. 

26. Speed Curve Plot- 
ting with Grades and 
Curves. — As an illustra- 
tion of the method of 
plotting speed curves over 
runs having grades and curves, consider the same car and 
equipment making a 0.9 mile run over a roadway the plan 
of which is shown in Fig. 32 ; all other conditions to remain 
unaltered. 



Fig. 32. 



68 TRACTION AND TRANSMISSION. 

As before, to produce an acceleration of 1.5 miles per 
hour per second on a level track requires 

24. "^2 

1.5 X 100 X =912 pounds per motor, 

4 

and to overcome train resistance 70 pounds additional 
must be exerted. But as the car must be accelerated on a 
2.3 % up grade, a further tractive effort must be exerted 
amounting to 

20 X 2.3 X 6.08 = 280 pounds per motor. 

This total force of 1262 pounds is produced when each 
motor takes 77 amperes, as obtained from Fig. 23, and this 
current value is maintained moderately uniform until the 
motors operate on the full line voltage of 600 volts, which 
occurs when the car has attained a speed of 15.3 miles per 

hour. The time required therefor is -^^ = 10.2 seconds, 

.1-5 
and the distance traversed during this interval with uni- 

I < ^ 
formly accelerated motion is -^^ X 10.2 X 1.467 = 114 feet. 

2 

These values constitute the first points respectively of the 
speed and distance curves for this particular run, and are 
shown at A and a' on the curves of Fig. 33. 

When the speed of the car has reached 18 miles per hour 
the total tractive effort exerted by each motor is 840 pounds. 
The grade resistance is still 280 pounds, but the train 
resistance at this speed is now 84 pounds per motor. There- 
fore the net tractive effort producing acceleration is 840 — 
(280 + 84) =476 pounds; whence the rate of acceleration 
at the instant the velocity of the car is 18 miles per hour is 

476 



100 X 6.08 



= 0.78 mile per hour per second. 



SPEED CURVES. 



69 



133J 



O 

o 
o 

10 



o 



o 
o 
o 



o 
o 
o 

CO 



o 
o 
o 



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o 
o 




























































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1 V 


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ynoH y3d S3nm 



70 



TRACTION AND TRANSMISSION. 



The time required for the car to gain this velocity incre- 
ment of 2.7 miles per hour is 

2.7 -^ i (1.5 + 0.78) = 2.36 seconds, 

and the space traversed during this interval is 

2.36 X i (15.3 + 18.0) X 1.467 = S7-S feet. 

Thus, 12.56 seconds after the car started from rest it 
acquired a speed of 18 miles per hour and covered a dis- 
tance of 171.5 feet. These values constitute second points 
respectively on the speed and distance curves, and are 
indicated at h and V in Fig. 33. Other points are similarly 
determined, as noted in the following table, the process 
being continued until a distance of 800 feet has been passed 
over by the car. At this place the grade ceases and the 
remainder of the run is on a level track. 



<6 


1 


H 


si 


0) 

% 


Si 




1 


C 4-> 

C/3 


^ q5 

-^ 


A 


15.3 

18 

20 

22 

24 

239 






1262 

476 

290 

154 
48 

53 


150 

0.78 

0.48 

0.25 

0.079 

0.088 


10. 2 

2.36 

3-i8 

5.48 

12.15 

II. 21 


10. 2 
12.56 

15-74 
21 .22 

32.43 


114. 
57.5 

88.6 
168.9 
409 
376 


114. 

171. 5 

260. 1 

429 
838 
805 


b 
c 
d 
e 


840 
660 
530 
430 

435 


84 
90 
96 
102 
102 



It is seen in the table that point e was corrected in order to 
approximate the distance of 800 feet more closely. 

Beyond the grade the net tractive effort for producing 
acceleration becomes larger by the amount of 280 pounds 
per motor, and thus the speed of the car increases more 
rapidly than before. Continuing the tabulation until the 



SPEED CURVES. 



71 



car strikes the curve, there obtains (compare with points 
e to h of table of § 24) the following: 





T3 


4J 

^1 




1 




.s-s 


1 






Ti 


CO 


> 






Pi; 




13 

e2 


1^ 

C/3 


1" 


f 


26 


360 


108 


252 


0.41S 


^^zs 


40.78 


305 


mo 


g 


28 


300 


115 


i«S 


0.304 


5-57 


46.35 


220 


1330 


k 


30 


255 


122 


133 


0.219 


7.6s 


54.0 


326 


1656 


t 


32 


220 


130 


90 


0.148 


10.90 


64.9 


495 


2151 


3 


34 


185 


139 


46 


0.072 


18.20 


83.1 


880 


3031 


31 


33-3 


198 


135 


03 


0.104 


10.30 


75.2 


493 


2644 



Since the car encounters a curve after running 2650 feet, 
a readjustment of point j of the speed curve was neces- 
sary, because after passing this place the rate of accele- 
ration of the car decreases since some tractive effort is 
required to neutralize the increased flange friction. Taking 
this as 0.5 pound per degree per ton for simplicity, the 
additional tractive effort is -^sV-X 6.08 X 0.5, or 36 pounds. 
The length of the curve is 480 tt -^ 2 = 754 feet; that is, 
the curve ends at a distance of 3404 feet from the starting 
point. The figures in the following table refer to the car 
movement on the curve of 480 feet radius. 





1 


^1 


a3 

is 
(-1 


1 




z 
is 


6 
I 




-d 
_ 


0. 




H^ 


^ 




H 


ex 


H 


k 


34 


185 


139 


10 


0.0165 


11.62 


86.82 


573 


3217 


I 


34.2 


181 


145 








24.22 


I I I . 04 


1212 


4429 



Had the curve extended over a greater distance the ulti- 
mate velocity of the car thereon would have been 34.2 miles 



72 TRACTION AND TRANSMISSION. 

per hour; but the curve ends before this velocity is acquired 
and thereafter the car runs on a tangent level track. The 
time when the car emerges from the curve is shown by the 
distance curve of Fig. 33, and the acceleration curve from 
this time on may now be completed along the lines previ- 
ously outlined. The braking and coasting curves are then 
drawn in their proper positions, so that the enclosed area 
truly represents a 0.9 mile run. The completed speed 
curve is shown as OABCDE in Fig. 33. 

By reference to this curve it is seen that the power is 
cut off from the car when its velocity is 32.1 miles per 
hour and when it has been running for 65.6 seconds. Dur- 
ing this time the car traveled 2175 feet, as indicated by 
the distance curve. While the car is coasting for 67.4 
seconds it passes over 

674 X i (32.1 + 17.9) X 1.467 = 2465 feet. 

Thus the brakes are applied when the car is distant 4640 feet 
from the starting point. The time required to bring the 
car to rest from a velocity of 17.9 miles per hour at the 
prescribed rate of braking is 8.95 seconds, and the distance 

1 7 o 
traveled during this period is 8.95 X -^-^ X 1.467 = 117 ft. 

2 

Thus the total length of the run as determined by summa- 
tion of the separate distances is 4757 feet, a value which 
exceeds the true length of run by but 5 feet. Distance 
curves therefore serve as admirable checks in the plotting 
of speed curves. 



SPEED CURVES. 73 



PROBLEMS. 

14. Plot a complete acceleration curve of a car weighing 20 tons with live 
load and equipped with two 50-horsepower, direct-current motors whose 
characteristic curves are given in Fig. 23. The initial acceleration rate is 
to be 1.3 miles per hour per second and the schedule speed is specified at 
15 miles per hour on a tangent level track. What is the maximum possi- 
ble velocity of this car on such a roadway? / 

15. Complete the speed curve of the equipment mentioned in problem 14 
over a |-mile level roadway, allowing a 15-second stop at the following sta- 
tion. The braking rate is specified at 1.5 miles per hour per second. 

16. What is the shortest running time that a motor car weighing 43 tons 
total with passengers and equipped with two 200-horsepower, 5 50- volt, 
direct-current motors whose characteristic curves are shown in Fig. 24, can 
complete a one-mile run up a uniform grade of 1.5 %? The acceleration 
and braking rates are 2 miles per hour per second. 

17. An 8-car New York Subway train having five motor cars each 
equipped with two 200-horsepower, 5 50- volt motors, weighs 320 tons in- 
cluding live load. The characteristic curves of the motors are shown in 
Fig. 24. Plot the acceleration portion of the speed curve for an initial 
acceleration of two miles per hour per second on a tangent level track. 

18. If the schedule speed of the train in the foregoing problem is 25 miles 
per hour and the rate of braking is 2^ miles per hour per second, com- 
plete the speed time curve of problem 17 for a run of ij miles, allowing a 
ten-second stop. 



74 TRACTION AND TRANSMISSION. 



CHAPTER V. 
RAILWAY MOTOR CONTROL. 

27. Direct-current ControL — The motor-control equip- 
ment of an electric car or train serves to regulate the 
speed and direction of rotation of the motors and to govern 
their action during periods of initial acceleration. The 
most important function of a railway motor controller is 
to maintain a sufficiently uniform change of velocity during 
initial acceleration, due consideration being given to the 
durability of the apparatus and to the comfort of passengers. 
Thus the variations in the starting current from the aver- 
age value necessary to produce the required tractive effort 
for the specified rate of acceleration must be so restricted 
that the accompanying fluctuations in torque will not be 
injurious to the equipment or unpleasant for the passengers, 
and the maximum current attained will not give rise to 
commutation difficulties. 

With direct-current series motors two general methods 
of control are in use: i, rheostatic control, and 2, series- 
parallel control. 

28. Rheostatic Method. — In the rheostatic method, 
for use with one or more motors, resistance is connected 
in series with the motor circuits, which is varied so as 
to regulate the voltage impressed upon the motors. A 
scheme of connection for a rheostatic railway controller 
is indicated in Fig. 34. Successive portions of this resist- 
ance are short-circuited by closing switches i, 2, 3, and 4. 



RAILWAY MOTOR CONTROL. 75 

in the order named, thus gradually increasing the pressure 
applied to the motor terminals. This method, although sim- 
ple, is infrequently employed because the loss in the regulat- 
ing resistance is not conducive to economical operation. 



f 1 2 3 4 

-O CH-pO O— r-O O-T-O O-i 



Fig. 34. 

29. Series-parallel Method. — The series-parallel method 
of railway motor control is extensively used for equipments 
with two (or any multiple of two) motors. The car is 
started from rest and accelerated by first placing the two 
motors and a resistance in series and then cutting out the 
resistance step by step until the motors are operating in 
series on full voltage. Since with all the resistance cut 
out there is no unnecessary PR loss, this is called a running 
connection, and the controlling mechanism is said to be 
on a running point. To increase the speed further, the 
motors are placed in parallel, with a resistance in series 
with both. This resistance is then cut out step by step 
until the motors are each operating on the full line voltage. 
This also constitutes a running connection. 

The circuits of a series-parallel controller are more com- 
plex than those of the rheostatic type, since additional con- 
nections are required to effect the transition from the series 
to the parallel position. For accomplishing this change 
three different methods may be used. Their distinctive 
features are respectively (i) the shunting or short-circuit- 
ing of one of the motors; (2) the opening of the power 



76 TRACTION AND TRANSMISSION. 

circuit; (3) the maintenance of full current through all 
motors during transition. 

Most of the so-called Type K controllers, ordinarily 
used with single-car equipments, operate according to the 
first method, the successive steps of which are essentially 
as follows: the starting resistance is gradually cut out until 
the motors operate in series on full line voltage ; thereafter 
a portion of the total starting resistance is reinserted in 
series with the two motors, one of which is then shunted 
or short-circuited, thus connecting the other motor across 
full voltage but with a protective resistance in circuit. 
The short-circuited motor is thereafter connected in paral- 
lel with the other, the resistance now being in series with 
both motors; this resistance is subsequently cut out in suc- 
cessive steps. 

The second method of series-parallel control, that of 
opening the power circuit during transition, exemplified 
by Type L controllers, is merely an extension of the first, 
intended for use with motors of very large capacity. This 
method is now rarely employed because of its inferiority to 
the third method, which has been developed to meet the 
same requirements more effectively. 

The third method of transfer from the series to the 
parallel position is used with multiple-unit control, and 
also applied to a few Type K controllers designed to meet 
the exacting conditions associated with large motor capacity 
and high voltage. During transition, full current is main- 
tained through all the motors by means of a ^^ bridge^' 
connection. A scheme of connections illustrating th's 
type of series-parallel control is shown in Fig. 35. The 
controller performs the following operations: switches A 
and B are closed, thus placing both motors and all the 



RAILWAY MOTOR CONTROL. 



77 



resistance in series between the trolley or third rail and 
ground. This connection, which corresponds to a slow speed 
that is suitable for switching in terminal yards, is passed 
over quickly when accelerating at the usual rate. The 
first movement of the controller handle accomplishes the 
simultaneous closing of switches 5, 6, and 7. Switches i 
to 4 are then closed consecutively, followed by the closing 
of switch C and the subsequent opening of switches 2 to 7 
and Bj thus connecting the motors in series across the line 



4§i: 



r. 



)0-' 



|-0( 




' On 



Fig. 35. 

through the ^^ bridging '^ switch C. Thereafter switches a 
and h are closed. Thus two currents will flow through 
switch C in opposite directions, one from the trolley through 
the motors to ground and the other through the resistance 
to ground. With properly proportioned resistances prac- 
tically no current wall pass through C, and consequently 
this ^^ bridging" switch may be opened, thereby placing the 
motors in parallel, with resistance in series with each. 
After this, switches 2 and 5, 3 and 6, and 4 and 7 are closed 
progressively, thus finally placing each motor on full volt- 
age. This method is desirable in that no motor is sub- 
jected to a sudden increase in voltage nor is the circuit 
opened at any time. Unnecessary variations in torque are 
therefore avoided. 



78 TRACTION AND TRANSMISSION. 

When four motors are installed on a car, they may first 
be connected in series, then each pair in parallel with the two 
groups in series, and finally all connected in parallel ; this is 
known as the series, series-parallel, parallel method. Usually, 
however, the motors are arranged in two groups, each con- 
sisting of two motors permanently connected in parallel and 
treated as a single unit in so far as their control is concerned. 

30. Starting Resistances. — The design of starting re- 
'sistances for use with railway controllers requires a knowl- 
edge of the allowable variation in torque during accelera- 
tion. When a motor is started from rest with resistance 
in series, the current gradually decreases with increase in 
speed because of the generation of more and more counter 
E.M.F,, until a portion of the resistance is cut out, caus- 
ing a sudden increase in current. Thereafter the current 
gradually decreases again with further increase in speed 
until another portion of the resistance is cut out, which 
causes a sudden rise in current as before. This current 
fluctuation continues until full line voltage is appHed to 
the motor terminals. These current variations produce 
corresponding variations in torque, which, if violent, cause 
unevenness in the velocity increase of the car, resulting in 
discomfort to passengers and in severe mechanical stresses 
on the apparatus. Experience shows that, in general, the 
maximum and minimum values of torque should not differ 
from the average value required to produce the prescribed 
acceleration by more than ten per cent of such average 
value. Since the iron of a direct-current series motor ap- 
proaches saturation when taking the large current required 
for starting, the torque exerted is approximately proportional 
to the current. Hence the current is restricted to a similar 
range of variation. 



RAILWAY MOTOR CONTROL. 79 

Fluctuations in the current supplied to a series motor 
affect its field strength and thus produce changes in the 
counter electromotive force generated, which must be 
considered in designing the controller resistances. The 
necessary information relative to these changes of counter 
E.M.F. is obtained from the saturation curve of the motor, 
a curve which shows the electromotive force generated 
in the armature as a function of the field (or armature) 
current when the machine is driven at constant speed. 
This curve is readily computed from the resistance of the 
motor and its characteristic curves. The electromotive 
forces corresponding to any given values of current evi- 
dently bear the same relation to each other whatever that 
constant speed may be. 

Rheostatic Controllers, The proper resistance units for 
a rheostatic railway controller may be determined as 
follows: ' 

Let E = line voltage, 

Rm = resistance of motor, 
rij ^2, ^3, . . . , r^ = the respective controller resistances 

in series with the motor when 
the controller arm is on contact 
studs I, 2, 3, . . . , 7^, Fig. 36. 

£2, £3, . . . , En = the respective counter electromotive 

forces generated at the instants 
when the arm makes contact 
with studs 2, 3, 4, . . . , fly 

£/, £2', . . . ,En = the respective counter electromotive 

forces generated at the instants 
when the arm breaks contact 
with studs I, 2, 3, . . . , w, 



8o 



TRACTION AND TRANSMISSION. 



/ = average current necessary to produce the required 
tractive effort for the prescribed rate of accel- 
eration, 
/max = maximum current value and /min = minimum cur- 
rent value as dictated by the allowable range 
of current variation, 




Fig. 36. 

£max = the electromotive force corresponding to the cur- 
rent /max as determined from the saturation 
curve. Fig. 37, 

-Emin = the electromotive force corresponding to the cur- 
rent /min, Fig. 37, 

and for convenience let 

-'-'mm 

and 



At the instant when the arm touches stud i, the resist- 
ance Yi should be such that the current flowing through 



the motor will not exceed I^ 



then 



WAY MOTOR CONTROL. 


8i 


T ^ 


(i) 


-I max — .. , r, ' 



whence the total resistance of the rheostat is 

E 



Ti = 



— Rm- 



(2) 




Fig. 37. 



As the motor starts from rest and accelerates, the current 
gradually decreases, and at the instant when it reaches the 
value /min the arm should leave stud i ; then 



ri + Rm 
Dividing (3) by (i) there results 

which when solved for Ei gives 



(3) 



82 TRACTION AND TRANSMISSION. 

E,' = E{i-K). (4) 

At the instant when the arm touches stud 2 the motor 
current should again be /max, which is now equal to 

T __ E - E2 , V 

whence 

r^=^-R,.-Y^- (6) 

^ max J- max 

Since £2 and £/ are generated at the same speed and 
with the respective field currents /max and /mm, reference 
to the saturation curve shows that 

£1 -fimin 

and therefore 

E2 = qEi\ 

which, by substitution from (4) , becomes 

E, = Eq{i-K). (7) 

At the instant when the current has again decreased to 

/min the arm leaves stud 2 and 

/min = ^ j_J • (8) 

Dividing (8) by (s), 

IT - E "" ^2^ 

^-£^^' 

from which 

£2' = E{i-K)+ KE2, 

whence by substitution from (7) 

£2' = E{i-K)+ EqK (i - K). (9) 

Proceeding in a similar manner there results 

E j^ Ez / \ 

^3 = 7 — — J<m — 7 — ' Vio; 



RAILWAY MOTOR CONTROL. 83 

£3 = qE^' = Eq(i-K) + Eq'K (i - K), (11) 

£/= £ {i-K)+ KEz = E{i-K) + EqK (i - K) 

+ Eq^K'{i -K), (12) 

and so on. 

The resistance of each of the various steps may now be 
determined; thus, subtracting (6) from (2) and substitut- 
ing from (7), the portion between studs i and 2 is 

r,-r, = ^^/-q{z-K). (13) 

-^ max J- n 



<- max ■*■ max 



Similarly, 

r2-n = -^{E,-E,) = ^{i-K)=qKin-r2), (14) 

J^ max J- max 



I .^ ^s_EfK} 



J- max -^ max 

and so on. 

An expression for the total number of steps required may 
be derived, but it is more convenient to proceed by first 
determining the total resistance by equation (2), then 
computing successive steps by equations (13), (14), etc., 
until the sum of the resistance steps thus obtained is approx- 
imately equal to (preferably equal to or greater than) the 
total resistance. This determines the number of steps into 
which the total resistance is to be divided. 

The foregoing equations may be used in designing the 
starting resistances of rheostatic controllers for any num- 
ber of motors, connected in any way, provided appro- 
priate values are substituted for /max and i?^. The same 
expressions may also be employed for calculating the series 
resistance steps of series-parallel controllers. 

Series-Parallel Controllers. The design of the parallel 
resistance steps for series-parallel controllers involves a de- 



84 TRACTION AND TRANSMISSION. 

termination of the proper resistances to be connected in 
series with a motor (or motors) already in operation and 
therefore generating a definite counter electromotive force. 
This is a more general problem of which the preceding deri- 
vation is a particular case. Thus, if the controller shown 
in Fig. 36 is to be placed in series with a motor that has 
already attained some definite speed because of its previ- 
ous operation in series with another motor, the equations 
governing the design of the rheostat must be modified as 
follows. 

At the instant when the lever arm touches stud i the 
current flowing is 

/max = , p^ > (16) 

Ti + Km 

where £1 is the counter electromotive force that is being 
generated at this instant. The other symbols retain their 
former significance. Herefrom 

n-^-Rr.-^- (17) 

■^ max ^ max 

At the instant when the arm leaves stud i the current 
flowing should be as before, 

/min = ^^'- (18) 

Dividing (18) by (16) and solving for Ei there results 

E,' = E{i-K)+KEi, (19) 

Again, at the instant when the arm touches stud 2 the 
current should again be 

r E — E2 / \ 

/max = T^-' (20) 

r2 + Rm 

consequently 



RAILWAY MOTOR CONTROL. 85 

^2 =y - Rm -J (21) 

■^ max -^ max 

As before, £2 = qEi, 

whence by substitution from (19) 

£2 = Eq{i -K) + qKEi. (22) 

The instant the arm leaves stud 2 the current diminishes to 



E-E2' 



(23) 



^2 + Rm 

Dividing (23) by (20) and solving for £2', 

£2' = £ (i - i^) + EqK (i-K)+ qK'Ei. (24) 
Herefrom 

E T, Es , V 

rz = j Rm- J — (25) 

-^ max -*■ max 

and 

£3 = ^£2' =^Eq(i-K)+ Eq'K (1 - K) + q'K'E,, (26) 
and so on. 

Proceeding exactly as in the foregoing derivation there 
are obtained the following expressions for the resistances 
of the various steps of the controller: 

n-r,= rp^{E,-E^)=^{x-K)-^(i-qK),{27) 

-'max 



'- max ■*■ max 



r2 



-r^=-~{Ez- £2) = qK (n - r^ , (28) 

^ max 

rz-r^= - — ^ (£4 - £3) = qK {r^ - rs), (29) 

-'max 

arid so on. 

If the controller having the resistance steps under con- 
sideration is to be used in starting a motor (or motors) 
from rest, £1 is equal to zero and the equations reduce to 
the forms previously derived. If, however, the motor is 
already in operation, £1 will have some value greater than 
7 



86 



TRACTION AND TRANSMISSION. 



zero. In calculating the parallel resistance steps of a 

series-parallel controller, this value may be determined 

from the fact that the total resistance for parallel operation 

should be of such magnitude as to allow the current to 

increase from Ijnin to /max when the motors are transferred 

from the series to the parallel connection. Herefrom it 

follows that ^ ^ 

El = qE,, 

where Es is the counter E.M.F. per motor at the instant 
when the series connection is interrupted. Since the total 
counter E.M.F. generated when the motors are running 
in series without resistance is equal to the line voltage 
minus the total resistance drop, the value of E^ may readily 
be obtained in any given case. Thus, for a two-motor 
equipment 

2 

A definite knowledge of the resistance of the motor con- 
nections and car wiring is conducive to even greater accu- 




FiS J3. 



racy in the determination of the controller resistance units. 
A three-point grid resistance manufactured by the 
Westinghouse Electric Company is shown in Fig. 38. 



RAILWAY MOTOR CONTROL. 



87 



31. Numerical Example. — As an illustration of the 
method of applying the foregoing equations to the calcu- 
lation of resistance steps, consider the design of a series- 
parallel drum-type controller for use on a car equipped 
with two 35-horsepower, 500-volt motors. The saturation 
curve of the motors is shown in Fig. 39, and the resistance 



30 



20 



10 

















^^ 














^ 










./^ 


^^ 












A 














/ 


/ 












/ 


/ 














/ 

















10 



20 



30 40 50 

AMPERES 

Fig. 39. 



60 



70 



80 



of each motor is 1.18 ohms. The operating conditions 
are such that an average starting current of 60 amperes 
per motor is necessary to produce the prescribed initial 
acceleration rate, and the controller specifications require 
that the limiting values of current shall not differ from this 
average value of 60 amperes by more than 10 %. 
In this problem 



^max = 60 X I.I = 66 amperes, 
^min = 60 X 0.9 = 54 amperes, 



and therefore 



88 TRACTION AND TRANSMISSION. 

S4 2.92 

X=g =0.818, 2=^=i.io. 

and qK = 1,1 X 0.818 = 0.90. 

The total starting resistance required for the operation 
of the two motors in series is 

ri = ^- — 2 X 1. 18 = 7.58 — 2.36 = 5.22 ohms, 
66 

and the various series resistance steps into which it is 
divided are: 

500 X I.I 7 o o\ -u 

Ti — r2 = - — — (i — 0.818) = 1.52 ohms, 

66 

r2 — rs = 0.9 X 1.52 = 1.37 ohms, 

f 3 — ^4 = 0.9 X 1.37 = 1.23 ohms, 
and 

Ta — r^ = 0.9 X 1.23 = I. II ohms, 

making a total of 5.23 ohms. 

The counter E.M.F. generated at the instant when the 
motors are placed in parallel is 

77 i>i (50 - 2 X 1. 18 X S4) 1^ 

El = ^^^ ^^^ = 205 volts. 

Hence the total resistance required for parallel operation is 

ri= -^ — — ——^ =^.79 — (o.so + i.SS) = 1.6; ohms, 

66X2 2 66X2 ^ '^ ^ ^^ ' ^^^ ^ 

and the various parallel resistance steps into which it 
should be divided are: 

,,-.,= 52^^ (,_ 0.8x8) - -^ (X ^ 0.9) 
66 X 2 66 X 2 



and 



= 0.758 — 0.155 = 0.603 ohms, 
^2 — ^3 = 0.9 X 0,603 = 0.543 ohms, 



RAILWAY MOTOR CONTROL. 



89 



n — r^ = 0.9 X 0.543 ~ 0.489 ohms, 

constituting a total of 1.635 ohms. 

32. Field Control. — With commutating-pole direct-cur- 
rent railway motors additional running speeds may be 
obtained by varying the number of turns on the main field 
without introducing commutation difficulties. Motors 
utilizing this method, called field-control motors, have more 
field turns than they would otherwise and have their field 
windings divided into two parts, so that both parts may be 
connected in series to secure high torque during starting 
and only one part is used during operation at high speed. 
The benefits derived in using such motors are the lowering 
of the starting current and the reduction of energy loss 
during the starting period. 



40 






\ 






No. 307-C4 
MOTOR 










\ 


\ 






GEAR RATIO 15:69 
33"WHEELS 






A 


30 




\ 


\ 












F 


>^ 


y 






y \ 


.... 










^ 


^.. 




20 




1 


%- 


S, 




A 


/4\ 


y 












\ 




>^ 


w\ 










10 












't^ 


'^""•-^ 




-— 










y 


^ 


y 


















^ 


^ 


y^ 

















3200 



2400 O 



> 

I- 



i6oo: 



25 



50 



75 
AMPERES 

Fig. 40. 



100 



125 



Fig. 40 shows the tractive effort and speed curves of a 
50-horsepower, 600-volt Westinghouse field-control motor 



90 TRACTION AND TRANSMISSION. 

for full field (F.F.) and short field (S.F.) conditions. If a 
car requires each motor to exert a tractive effort of 1600 
pounds at starting, the motor will take 81 amperes on full 
field until a speed of 124 miles per hour is reached. Then 
if the short-field connection is made this same tractive 
effort would demand 90 amperes until a speed of 14.2 miles 
per hour is attained. Thus the motor without field control 
(that is, the equivalent of short-field connection) would take 
90 amperes for the starting time from o to 14.2 miles per 
hour, whereas the field-control motor would only take 81 
amperes for 7/8 of that starting time. In consequence less 
energy is taken with such motors than with others of 
equivalent rating, the saving in energy having been found 
to be from 5 to 15 per cent in various service tests. 

33. Alternating-current Control. — Single-phase series 
motors in railway service are controlled, like direct-current 
motors of the series type, by varying the pressure applied 
to their terminals. This variation may be effected by the 
standard direct-current method previously described. More 
efficient means of potential regulation are, however, avail- 
able with alternating current, so that the large PR loss 
incident to the use of starting resistances may be avoided 
and a greater number of running points obtained. Two 
general methods of control peculiar to alternating currents 
are at present employed with single-phase equipments: i, the 
induction regulator, and 2, the compensator method. 

Induction Regulators, — In starting a car or train by the 
former method the voltage impressed upon the motor 
terminals is gradually increased by means of a single-phase 
induction regulator. This device is essentially a trans- 
former of which one coil is movable with respect to the 
other, the windings being arranged in a manner similar 



RAILWAY MOTOR CONTROL 



91 



to those of a coil-wound induction motor. The primary 
coil is usually connected to suitable taps on an autotrans- 
former or compensator, used to step down the voltage. 
The secondary coil is placed in series with the motor cir- 
cuit, which is likewise connected to transformer taps of 
suitable potential. By changing the relative position of 
the regulator coils the effective E.M,F, induced in the 
secondary winding of the regulator may be varied from 
zero to a definite maximum value in either direction, that 
is, in phase with or in phase opposition to the E.M.F. 
impressed upon the motor circuit by means of the trans- 
former. Thus, if Et is the E.M.F, between the transformer 




GROUND 



Fig. 41. 



taps to which the motor circuit is connected, and Er is 
the maximum E.M.F. induced in the secondary coil of the 
regulator, then, neglecting the impedance drop in the 
wiring, the pressure applied to the motors may be varied 



92 



TRACTION AND TRANSMISSION. 



through all values from Et — Er to Et + Er according to 
the cosine of the angle of displacement between the axes 
of the two windings. This method of control is illustrated 
by the scheme of connections shown in Fig. 41, where C 
is the autotransformer which is connected across the Hne, 
S is the secondary coil of the induction regulator and is in 
series with the motor circuit, and P is the primary coil 
thereof, which in this case is the movable element of the 
regulator. Evidently every possible position of the control- 
ler will result in a definite voltage upon the motors, so this 



2 O — ^15W^ 
P 

1 O 



TO MOTORS 




GROUND 



Fig. 42. 



method of control yields a multiplicity of running positions. 
The large weight and low power factor of the regulators, and 



RAILWAY MOTOR CONTROL. 93 

the complicated mechanism required for their operation 
are, however, serious objections which tend to retard the 
further adoption of this type of control. 

34. Compensators. — In the compensator method of 
control the voltage at the motors is regulated by varying 
the ratio of transformation of a compensator, which serves 
also as a step-down transformer in those installations where 
high trolley potentials are used. One terminal of the motor 
circuit is connected to ground. The other terminal may be 
successively connected to a series of compensator taps so 
arranged that during initial acceleration the E.M,F. applied 
to the motor circuit may be increased in suitable steps 
until each motor operates on rated voltage. 

The connections of a compensator-type controller should 
be such that the transition from one compensator tap to 
another may be effected without interrupting the motor 
current or short-circuiting any portion of the compensator 
winding. For example, in transferring the motor connec- 
tion from tap i to tap 2 of the compensator C, shown in 
Fig. 42, an uninterrupted flow of current through the 
motors is maintained by closing switch 2 before switch i is 
opened. In order that this procedure may not short- 
circuit the portion of the compensator winding included 
between taps i and 2, a preventive coil P is connected in 
series with switch 2 as shown. The resistance R and the 
reactance X of this preventive coil are so proportioned that 
the impedance drop Z7, resulting from the passage of the 
motor current /, is equal in magnitude and opposite in 
phase to the voltage E existing between taps i and 2 of the 
compensator winding. This relation is indicated by the 
vector diagram in Fig. 42, where is the angle by which 



94 TRACTION AND TRANSMISSION. 

the motor current lags behind the pressure £, which is of 
course in phase with the voltage impressed upon the motor 
circuit by means of the compensator. It is evident from 
this figure that the values of resistance and reactance 
required depend on the power factor, cos 0, of the motor 
circuit. Since the power factor varies through a consider- 
able range during the period of uniform acceleration, it is 
desirable to connect in series with each compensator switch 
a preventive coil designed to meet the particular conditions 
obtaining at the instant when that switch is closed. This 
method of control has, however, the disadvantage of requir- 
ing a relatively large number of preventive coils no two of 
which have the same constants, yet each must be designed 
to carry the full motor current. 

In the so-called multiple-switch method of compensator 
control, now extensively employed, the preventive coils 
are used as auto-transformers to divide the motor current 
between two or more compensator switches. Thus, at 
each running point of the controller the motor circuit is 
connected to a set of two or more successive compensator 
taps, each of which supplies a definite fractional part of the 
motor current. The essential features of this method are 
illustrated in Fig. 43. In the particular scheme of connec- 
tions there depicted, three preventive coils are used to 
divide the motor current into four approximately equal 
parts. The first running position of the controller is at- 
tained by closing switches i, 2, 3, and 4. The voltage 
applied to the motor circuit when the controller is in this 
position is evidently equal to the potential relative to 
ground of a point on the compensator winding midway 
between taps i and 4. When the controller handle is 
moved to the second running position switch i is opened, 



RAILWAY MOTOR CONTROL. 



95 



followed by the closing of switch 5. Similarly, to pass to 
the third running point, switch 2 is opened and then switch 
6 is closed; and so on until the motors are supplied with 
current at rated voltage through switches 5, 6, 7, and 8. 
It is obvious that during transition from one running point 




GROUND 



Fig. 43. 



to another the full motor current is maintained without 
short-circuiting any portion of the compensator winding. 
Since each switch is required to handle only a fractional 
part of the total current supphed to the motor circuit, this 
method is well suited for use with railway equipments of 
large capacity. 

In cases where single-phase series motors are required to 



96 TRACTION AND TRANSMISSION. 

operate on direct current over a portion of the roadway, 
some form of rheostatic or series-parallel control must be 
installed for use during the periods of direct-current oper- 
ation. The losses that would result from the use of start- 
ing resistances during the intervals of alternating-current 
operation are, however, in general sufficient to justify the 
installation of compensator control for use on the sec- 
tions where alternating current is employed. This com- 
pensator may constitute a part of the autotransformer 
which is used to step down the high trolley voltage asso- 
ciated with alternating-current traction to a lower value 
which is suitable for motor operation. The use of com- 
pensator control on road sections supplied with alternating 
current therefore involves little additional expense. 

35. Induction Motor Control. — The methods of control 
required with three-phase induction motors are essentially 
different from those employed with alternating-current rail- 
way motors of the series type. The latter methods are not 
applicable to induction motors in railway service, since 
the reduction in impressed voltage necessary in starting 
by any of these methods causes a prohibitive decrease in 
the capacity of such machines. The following methods are, 
however, available for the control of three-phase induction 
motor equipments: (a) variable resistances in the second- 
ary circuits of the motors; (b) changing the number of 
poles of the motors; (c) cascade operation of the motors. 

(a) Variable Resistance Method. The insertion of vari- 
able external resistances in series with each phase of the 
secondary windings of the motors by means of suitable 
slip rings constitutes the principal method of maintaining 
an approximately uniform torque during the periods of 
initial acceleration. These resistances are so proportioned 



RAILWAY MOTOR CONTROL. 97 

that the motor exerts at starting a torque sufficient for the 
prescribed acceleration rate. As the speed of the motor 
increases, causing a decrease in the E.M.F. induced in 
the rotor windings, the external resistances are cut out 
successively, thereby maintaining a moderately constant 
secondary current and thus uniformly increasing the speed 
at which the motor exerts the definite torque required. 
While this method possesses the advantage of simplicity, 
it does not permit of efficient acceleration because of the 
PR losses in the rotor resistances. It also provides for 
only one efficient running speed, since the induction motor 
is practically a constant-speed machine, the slip rarely 
exceeding 10 % of the synchronous speed which the motor 
closely approaches when the car runs at its ultimate veloc- 
ity on a level roadway. It is therefore desirable to employ 
in connection with this resistance method of control some 
means of changing the synchronous speed of the motors, 
thereby reducing the PR losses during acceleration and 
providing for one or more additional running speeds. 

(b) Variable MuUipolarity Method, In the second method 
of control the synchronous speed of the motors is varied by 
changing the number of motor field poles. If the frequency 
of the voltage be/ cycles per second, the synchronous speed 
in revolutions per minute is 

V =^, 

P 

where p is the number of pairs of poles on the induction 
motor. 

In order to change the number of poles of a given induc- 
tion motor it is necessary either to provide two or more 
separate windings, §ach of which is designed to yield a 



98 



TRACTION AND TRANSMISSION. 



different number of poles, or to employ a single winding so 
arranged that the number of poles which it produces may 
be altered by a suitable change in the connections between 
the various parts of the winding and the three-phase line. 
The latter method is the more desirable since no inductors 
are idle during operation. 

A simple arrangement of windings for carrying out this 
method is illustrated in Fig. 44, which shows the stator 
winding of one phase of an 8-pole — 4-pole, three-phase in- 
duction motor. The complete phase winding 1-3 is divided 



N 



N 
S 



N 



[8 POLES] 

[4 poles] 



\/ 



\ 



o 

1 



y 



o 
3 



/ 



\ 



6 
2 

Fig. 44. 



into the two parts 1-2 and 2-3 by a tap 2 at the middle 
point of the winding. Terminals i and 3 connect with the 
windings of the two other phases, which for clearness are 
not shown in this figure. The winding shown in Fig. 44 
differs from the usual induction motor winding in that only 
alternate poles are wound. To produce an 8-pole magnetic 
field the windings 2-1 and 2-3 are placed in parallel with each 
other by connecting tap 2 to one of the line wires and taps 
I and 3 to the neutral point of the phase windings. The 
coils are so arranged that when thus connected they pro- 



RAILWAY MOTOR CONTROL. 



99 



(luce poles which are all of the same polarity. Interme- 
diate poles of opposite polarity will therefore be formed 
between them, thus producing an 8-pole field as indicated. 
If, however, a 4-pole field is desired, windings 1-2 and 
2-3 are placed in series by connecting terminals i and 3 to 

2f— X 



POLES POLES 



Lj: 




O- 



-GROUND 

Fig. 45. 



line wires of the three-phase supply. One of the windings 
is thereby reversed with respect to the other and conse- 
quently the poles pertaining thereto will be of opposite 
polarity. The intermediate poles will then disappear, re- 
sulting in a 4-pole field. Fig. 45 shows the schematic ar- 
rangement and the controller connections for simultaneously 
changing the number of poles of all three stator phases. 



lOO TRACTION AND TRANSMISSION. 

(c) Cascade Method. The third method of three-phase 
induction motor control consists in operating two motors 
in cascade. In the cascade connection, or concatenation, 
of two induction motors, the rotors of both machines are 
mounted on the same shaft or otherwise mechanically 
coupled as by gears or connecting rods. The primary of 
the first motor is connected to the line and its secondary 
is connected to the primary of the second motor. The 
secondary windings of the latter machine are short-cir- 
cuited through suitable starting resistances. 

When two induction motors are started in cascade con- 
nection, the power output of the first machine consists in 
part of mechanical power delivered to the rotor shaft and 
in part of electrical power supplied to the primary of the 
second machine. During initial acceleration, the torque 
exerted by such a cascade set is maintained approximately 
constant by progressively cutting out the starting resist- 
ances. Thereafter the torque decreases with further in- 
crease in speed, approaching zero as the slip of the second 
motor decreases toward zero. Thus two motors connected 
in cascade approach, when operating under light loads, a 
definite limiting speed, which may be determined as follows: 
Let / = the frequency of the line E.M.F., 

Vi = the synchronous speed of the first motor in rev. 

per min., 
¥2= the synchronous speed of the second motor in 

rev. per min., 
V = speed of rotor shaft in rev. per min., 
pi = number of pairs of poles of the first motor, 
p2 = number of pairs of poles of the second motor, 
^1 = slip of the first motor, 
^2 = slip of the second motor. 



RAILWAY MOTOR CONTROL. 1 01 

Then 

F. = ^ (I) 
and 



7,= 



60 5i/ ^ 60/ / Fi - V \ ^ 60// _ V\^ 

Pt p2 \ Vi I p2 V Vj' 



which by substitution from equation (i) becomes 

60/ - PiV 

Vi - 7 (2) 

p2 

Since 

F2- F 

'^=~Fr~' 

therefore 

F=F2(i-52). (3) 

Substituting in equation (3) the value of V2 given in equa- 
tion (2), there results 

F = ^5l^(,_,^)^ (4) 

which shows that as s^ approaches zero V approaches the 
limiting speed, 

60/ 

Pl + p2 

Hence the synchronous speed of the two motors connected 

in direct concatenation is the same as that of a single 

motor having pi + p2 pairs of poles. 

Two similar induction motors connected in cascade 

share the load with approximate equality; thus the second 

motor utilizes a considerable portion of the energy that 

would otherwise be consumed in the starting resistances 

when operating at speeds below the synchronous speed of 

the combination. At the latter speed, however, the torque 

exerted is zero, and with further increase in speed, such as 
8 



102 



TRACTION AND TRANSMISSION. 



occasioned by running down grades, the torque becomes 
negative and the cascade set operates as a generator, return- 
ing energy to the line. 

In most cases of cascade control the motors are divided 
into groups, each of which consists of a main motor and an 
auxiliary motor, the latter being employed during cascade 

7 



TROLLEYS 




Fig. 46. 

operation only. In starting, each auxiliary motor is con- 
nected in cascade with the corresponding main motor, 
and the starting resistances in the secondary circuits of 
the former are cut out in successive steps. The cascade 
connection is then broken by short-circuiting the second- 
ary windings of the main • motor through the starting 
resistances, which are thereafter cut out progressively as 
before. Thus the auxiliary motors are required to operate 
only intermittently on a low voltage, and the full-speed 
power factor of the main motors is higher than would be 
the case if their load were shared with the auxiliary motors 
by connecting the latter across the line. Fig. 46 shows 
a scheme of connections for this method of control. 



RAILWAY MOTOR CONTROL. 103 

36. Controllers. — All types of railway motor control 
must include means for changing the direction of rotation 
of the motors. A series motor is reversed by interchang- 
ing the connections of either its field or its armature wind- 
ings. The standard method of reversing series motors 
which have no commutating poles is to reverse the arma- 
ture, while that of commutating-pole motors is to reverse 
the main field so that the relative directions of current 
through the armature and commutating-pole windings will 
be unchanged. With a three-phase induction motor 
reversal of direction is obtained by exchanging the con- 
nections of any two of the three leads that supply the motor 
with current. 

Hand Control. — The manipulation of the switches is ac- 
complished directly by hand or through the intervention 
of an auxiliary control. In the former system a motorman 
makes the necessary electrical connections by moving a 
handle at the top of a controller on the car platform. The 
movement of this handle causes the rotation of a vertical 
cylinder and thus permits of the successive connection 
of various contact studs thereon with stationary fingers, 
which, by means of suitable car wiring, are properly con- 
nected to the trolley or third rail, to the motors, and 
to the different rheostat terminals or compensator taps. 
Fig. 47 shows a Westinghouse controller, for series-parallel 
operation, with the cover removed. It has seven control- 
ling points in the series position and six in the parallel 
position, and the motors are short-circuited during the 
transition period. The direction of rotation of the motors 
is changed by moving a reversing lever and thus actuating 
a smaller cylinder which is m.ounted beside the main cylin- 



104 



TRACTION AND TRANSMISSION. 



der of the controller and is provided with suitable contact 
pieces for effecting the necessary change in connections. In-' 
terlocking devices are supplied, so that the reversing handle 
cannot be moved unless the controlling handle * is in such 
a position that connection with the trolley or third rail is 




Fig. 47. 

broken. The controlling handle also cannot be moved if 
the reversing handle is not properly set either for forward 
or backward motion of the car. The reversing handle can 
be removed from the controller only when in its neutral 
or ^^off" position, to which it cannot be turned unless the 
controlling handle is also in its ^^off " position, thus entirely 
disconnecting the motor circuits from the trolley or third 



RAILWAY MOTOR CONTROL. 105 

rail. Cut-out switches are provided, so that a defective 
motor or group of motors may be disconnected without 
interfering with the operation of the remaining motor or 
motors. As serious arcs are Hable to ensue upon breaking 
a circuit of 500 volts, the contact pieces and fingers are 
separated from adjacent ones by strips of insulating mate- 
rials, which are usually fastened to the inside of a separate 
cover. Such arcs are effectively disrupted by the field of 
an electromagnet, which is an essential part of controllers 
used with motors of large capacity. 

Multiple-Unit Control. — The system of motor control in 
which the switches are operated electrically or pneumatically 
through the intervention of an auxiliary circuit is called the 
multiple-unit system, since it is designed for the operation 
of several motor cars coupled together in a train, all the 
motors being controlled simultaneously from any master 
controller on the train. This system is now extensively 
employed not only for the operation of trains made up of 
motor cars and trailers but also for the control of electric 
locomotives and single-car equipments of large capacity. 
The control apparatus for each motor car or locomotive 
consists of a motor controller and two master controllers. 

The motor controller is composed of a number of switches 
or contactors, which close and open the various motor, 
resistance, or compensator circuits, and in general effect 
the changes in connection necessary in controlling the 
particular type of motor employed. Each of these con- 
tactors opens in a strong magnetic field, so that all arcs are 
immediately disrupted. A separate reversing switch gov- 
erns the direction of rotation of the motors. On motor cars 
all this apparatus is usually placed underneath the car, 
but on locomotives it is located in the cab. The contac- 



I06 TRACTION AND TRANSMISSION. 

tors and reverser may be operated by solenoids or by the 
use of compressed air controlled by electrically operated 
valves. In either case the solenoids or other electromag- 
nets that govern the movement of the switches are connected 
to the wires of the auxiliary circuit and are supplied with 
current in proper sequence by the hand-operated master 
controller. 

The master controller is considerably smaller than the 
ordinary street-car controller, but is similar in appearance 
and method of operation. The contact fingers of each 
master controller are connected to the wires of the auxiliary 
or control circuit, which usually consists of a multiple- 
conductor cable. By means of suitable couplers this con- 
trol cable is made continuous throughout any number of 
motor cars or locomotives operated together in a train. 
Current for the master control is taken from the line, or 
from a storage battery, through whichever master controller 
the motorman operates. Since this current is used solely 
for energizing the operating coils of the motor contactors, 
its value is comparatively small, usually not exceeding 
2.5 amperes for each car equipment. As the operating 
coils of each motor controller are connected to the wires 
of the control cable, any master controller on the train 
will simultaneously operate corresponding contactors on 
all the motor cars and thus establish similar motor con- 
nections on them. To avoid accidents which may occur 
through the physical disability of a motorman, the operat- 
ing handle of the master controller is sometimes provided 
with a button which must be held down in order to keep 
the auxiliary control circuit closed. In some cases the con- 
nections are so arranged that releasing this button applies 
the air brakes as well as opens the control circuits. 



RAILWAY MOTOR CONTROL. 



107 



The essential features of the multiple-unit system of 
control as applied to direct-current equipments are illus- 
trated in Fig. 48, which shows the^ principal motor and 
control circuits for one motor car. For clearness the re- 
verser is omitted, as are also the circuits necessary for its 
control. Assuming therefore that the reverser is properly 
set, the subsequent operation of the control system during 
initial acceleration is as follows: turning one of the master 
controllers to the first notch results in the closing of contac- 




Fig. 48. 

tors a, b, and //, due to current received from train wires i, 
2, and 8, thus establishing connection with the line and 
placing the two motors and a protecting resistance in 
series. Turning the master-controller handle successively to 
notches 2, 3, and 4 closes contactors c, d, and e, respectively, 
thereby progressively reducing the resistance by placing 
additional resistance units in parallel. When the controller 
handle is moved to the fifth notch, contactor / is closed, 



Io8 TRACTION AND TRANSMISSION. 

short-circuiting the resistances and connecting the motors 
in series across the Hne. In passing over the sixth or tran- 
sition notch contactors ^ to / and h are opened, followed by 
the closing of contactors g and i. This places the motors 
in parallel, with resistance in series with both. Turning 
the master-controller handle successively to notches 7, 8, 
9, and 10 progressively reduces the resistance as before 
until each motor is operating on full line voltage. 

The operation of the switches of a multiple-unit equip- 
ment in other than their proper sequence is prevented by 
various interlocking devices. For example, the connec- 
tions are so arranged that the reverser on a car cannot be 
actuated save when the contactors on that car are open, nor 
can the operating coils of the contactors be energized unless 
the reverser is properly set for the direction of motion indi- 
cated by the master controller. By means of a suitable 
cut-out switch the operating coils of the motor controller 
on any car can be disconnected from the control circuit 
without interfering with the operation of the train from 
either of the master controllers on that car. 

In multiple-unit equipments similar to that illustrated 
in Fig. 48 the progressive closing of the contactors is 
accomplished by turning the master-controller handle to 
successive notches. The maintenance of an approximately 
constant current during initial acceleration is therefore 
entirely dependent on the motorman's care and skill. It 
is often desirable to have the progressive operation of the 
contactors regulated by the motor current itself, in order 
that the variations in this current from the average value 
required during acceleration may be automatically re- 
stricted to the prescribed range, thereby insuring a uniform 
rate of acceleration and permitting the motorman to con- 



RAILWAY MOTOR CONTROL. 109 

fine his attention to the track and signals. This auto- 
matic acceleration is effected by means of current-limit 
relays having coils connected in series with the motor cir- 
cuit. Such relays may be arranged to regulate the pro- 
gressive closing of the motor-controller switches in either 
of two ways: i, by governing the movement of the master- 
controller contact cylinder, or 2, by governing the supply of 
current to the operating coils of the individual contactors. 

In the former method the contact cylinder of each master 
controller is connected to its operating handle through a 
helical spring. The cylinder is restrained by a magnetic 
clutch actuated by a current relay in series with the motor 
circuit. This relay is so adjusted as to release the clutch 
and allow the contact cylinder to advance one step whenever 
the motor current falls to its minimum limiting value. The 
master-controller handle may therefore be turned at once 
to any desired position, and the contact cylinder will 
follow in successive steps automatically governed by the 
motor current of the car on which the motorman is stationed. 
Evidently this method cannot be expected to give satis- 
factory results in cases where there is a material difference 
in the motor characteristics or the current requirements of 
the various cars composing a train. 

In the second method of automatic acceleration each 
motor car is provided with a current-limit relay that is 
designed and adjusted with reference to the requirements 
of that particular car equipment. The motor connection 
ultimately established on all the motor cars in a train is 
determined by the position to which the handle of the 
master controller is turned; but the successive steps neces- 
sary to attain this connection are governed independently 
for each car by the motor current of that car. The connec- 



no TRACTION AND TRANSMISSION. 

tions between the operating coils of the contactors and the 
control circuit are made automatically through auxiliary 
contacts on the contactors themselves; and the control 
current for closing these switches passes through the con- 
tacts of the current-limit relay. 

PROBLEMS. 

19. Determine the resistance units of a rheostatic railway controller for 
use with one 35-horsepower, 500- volt, direct-current motor having a resist- 
ance of 1. 18 ohms. The saturation curve of the motor is shown in Fig. 39. 
The average current required during initial acceleration is 50 amperes; and 
the maximum and minimum values of the current must not differ from this 
average value by more than 9 %. 

20. Determine the parallel resistance steps of a series-parallel railway 
controller for use with two 35-horsepower, 500-volt, direct-current motors, 
the saturation curves of which are shown in Fig. 39, the resistance of each 
motor being 1.18 ohms. An average current of 50 amperes per motor is 
required during uniform acceleration, and the limiting values of current 
are specified at 45 and 55 amperes. 

21. A 2 20- volt, single-phase motor is to be started by means of an induc- 
tion regulator with an initial voltage of 150. What are the angular dis- 
placements between the two regulator coils if 7 steps were required which 
yield equal voltage increments on the motor ? 

22. Determine the resistance and the inductance of a preventive coil to 
be connected in series with a certain compensator switch in order to effect 
sparkless transition by the method of control illustrated in Fig. 41. At 
the instant during acceleration when this particular switch is to be closed 
the 25-cycle motors have attained a speed such that the power factor of 
the motor circuit is 53 %. The motor current during the period of initial 
acceleration is approximately constant at 100 amperes and the E.M.F. 
between adjacent compensator taps is 25 volts. 

23. A motor car is equipped with four three-phase, four-pole induction 
motors arranged in pairs for cascade control. Each main motor has 5 
stator slots per pole per phase and 18 conductors per primary slot. Each 
auxiliary motor has 4 stator slots per pole per phase and 4 conductors per 
primary slot. Determine the equivalent number of stator conductors per 
pole when the motors are operating in cascade. 



ENERGY CONSUMPTION. Ill 



CHAPTER VI. 
ENERGY CONSUMPTION. 

37. Current Curves. — During the period of initial accel- 
eration of a car the current taken by the direct-current 
motors is maintained roughly constant by the control 
equipment, provided no changes of grade or curvature 
occur during this interval. Thereafter, until the car attains 
its ultimate uniform velocity on the particular roadway 
under consideration, the motor current decreases, at first 
rapidly and later more slowly. The instantaneous values 
of current from the time all the controller resistance is 
cut out until the power is shut off may be read directly 
from the performance curves of the motor, since each motor 
takes a definite current at the various speed values of the 
car during this period. A curve showing these instantane- 
ous current values in terms of time over a run is called a 
current curve of the railway motor, and serves as the basis 
for determining whether the assumed motor for a proposed 
installation can perform the prescribed service without 
overheating. 

It is usual to construct the curve of current per car 
rather than the current per motor in determining the 
energy consumption of a tentative equipment. When 
starting the car the two motors of a two-motor direct- 
current equipment are connected in series, or the four 
motors of a four-motor equipment, arranged for the usual 
series-parallel control, are connected in two groups joined 



112 TRACTION AND TRANSMISSION. 

in series, each group consisting of two motors connected 
in parallel. Four-motor equipments adapted for series, 
series-parallel, parallel control are not frequently employed. 
Hence from the instant of starting until the controller 
leaves the series position and connects all the motors in 
parallel with resistance across line voltage the current per 
car is equal to the current per motor times one-half the 
number of motors comprising the car equipment. At the 
end of this period, that is, when the motors are operating 
on the series position without resistance, the speed of the 
car is 



E-IR 



where E is the line voltage, / is the current traversing the 
motor and R is its resistance, and Vi is the car speed when 
the controller is full ^^on." It is at this speed that the 
current per car increases from its former value to the 
product of the current per motor times the number of 
motors on the car. While the motors operate on reduced 
voltage in the parallel position their current intake is con- 
stant, but thereafter the current per motor and that per car 
decrease as dictated by the motor performance curves on 
full line voltage. When coasting begins the current intake 
ceases and the current curve drops to zero. 

38. Average and Effective Currents. — The average cur- 
rent taken by the car over a complete run is based not 
merely upon the time during which the car receives power 
for propulsion nor upon the running time, but upon the 
time of the entire run including stops. This average 
current is determined by finding the area of the current 



ENERGY CONSUMPTION. II3 

curve and dividing it by the time of the run as given by 
the specified schedule speed. 

The current per motor which when flowing continuously 
will yield the same average copper loss in the windings is 
called the effective motor current and is equal to the square 
root of the average of the squares of the instantaneous 
current values. The effective current may be found by 
squaring a suitable number of values of the motor current 
and plotting these squared values on the time axis. The 
square root of the average ordinate of the curve drawn 
through these points and taken over the total time of run 
represents the equivalent motor current to which the heat- 
ing of the machine is proportional. 

39. Numerical Example. — As an illustration, consider 
a car equipped with four 50-horsepowerj 600-volt, G.E. 
2 1 6- A direct-current motors whose characteristic curves 
are shown in Fig. 23. The speed curve of this car over an 
0.8 mile run on a straight level track at a schedule speed of 
20 miles per hour is shown in Fig. 31 , which permits of a 
20-second stop. Determine (i) the average current intake 
for the car and (2) the effective current per motor. 

The current consumed by the motor as the car is accel- 
erated uniformly at 1.5 miles per hour per second from 
standstill to a speed of 16.9 miles per hour (see page 61) 
is maintained roughly constant at a mean value of 64 
amperes, the time necessary for the acquirement of this 
speed being 11.3 seconds. The current curve over this 
period will have a series of peaks occasioned by the vari- 
ations in voltage which is impressed upon the motors by 
the controller, but the exact shape of this part of the curve 
is of no particular consequence, and it may be drawn 
straight through the mean current value. Taking the 



114 



TRACTION AND TRANSMISSION. 



resistance of each motor as 0.30 ohm, the resistance drop 
thereof is 19.2 volts. Therefore the speed of the car at 
the instant when the transition from the series to the 
parallel position is made is 
600 



— 19.2 



600 — 19.2 



X 16.9 = 8.2 miles per hour. 



8.2 



This speed is attained in -^ = 5.46 seconds from the instant 

of starting. Thus, when the car is in motion for 5.46 seconds 
the current per car increases from 64 X I or 128 amperes 
to 64 X 4 or 256 amperes. The latter current value per- 
sists for 1 1.3 — 5.46 or 5.84 seconds. The current curve 
for the car before the motors operate on full line voltage is 
shown by OABCD in Fig. 49. 

Beyond the. point D the current curve is entirely depen- 
dent upon the motor performance curves, since the current 
intake per motor at different car speeds is directly obtain- 
able therefrom. The times at which these speeds obtain 
are given by the speed curve for the run under consider- 
ation. Thus the curve of current per car may be plotted 
in terms of time, as done herewith from the following com- 
putations : 



Speed of car 
(miles per hour). 


Current per motor 
(amperes). 


Current per car 
(amperes). 


Time of speed acquire- 
ment (seconds). See 
table, page 64. 


20 


48.2 


192.8 


13 84 


22 


42.1 


168.4 


16.26 


24 


37-4 


149 -6 


19 -45 


26 


33-9 


-^^s-^ 


23 65 


28 


310 


124.0 


29. 22 


30 


28.4 


113. 6 


36.88 


32 


26.3 


105. 2 


47.78 



ENERGY CONSUMPTION. 



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Il6 TRACTION AND TRANSMISSION, 

After 50 seconds coasting begins and the current curve 
is completed by drawing the vertical Hne EF. 

The area of the current curve per car is 7350 ampere- 
seconds, which when divided by the time of the run, namely 
144 seconds, gives the average current per car over the 
given run as 51.0 amperes. 

The curve of current per motor is shown in Fig. 50, 
as OABCD, the portion BC being also plotted from the 
values recorded in the foregoing table. The ordinates ol 
this curve when squared yield the curve OEFGD, the ar<^ 
of which is 90,930 amperes-seconds. The mean squa 
current over the given run which requires 144 seconds f 
its completion is 631 (amperes)^. Therefore the effecti 
heating current of the motor is 25.1 amperes. 

40. Efifective Motor Current for a Trip. — The effecti 
motor current for a trip over an entire roadway which 
divided into a number of individual runs distributed o^ 
several territorial sections on which different service con 
tions exist is obtained by averaging the squared cum 
values over all the runs and extracting the square root 
this average. Thus, for example, if the effective mo 
current values on typical runs on the city, suburban, i 
interurban sections of a certain railway are respectiv 
40 amperes for 25 minutes, 35 amperes for 20 minu 
and 28 amperes for 15 minutes, then the effective curn 
for the entire trip is | 



i 



/ 



(40' X 25) + G.^' X 20) + (28' X 15) 

25 + 20 + 15 



f 



v/^ 



_ .7 40,000 + 24,500 + 11,760 _ . 
~ V 60 " ^^ ' 



ENERGY CONSUMPTION. 



117 



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Il8 TRACTION AND TRANSMISSION. 

since this current flowing for 60 minutes would produce the 
same heating of the motor as is developed under the actual 
service conditions. 

41. Voltage Curve. — The line voltage in calculations 
of motor capacity is assumed constant and to have the 
same value everywhere on the roadway. The voltage on 
the motor equipment is thus considered constant, and a 
voltage curve would be a straight horizontal line, as OMNF 
in Fig. 49. The voltage impressed upon the motor 
terminals during the period of initial acceleration is 
increased from zero to full line voltage in steps by means 
of the controller. As a rule there are more than seven 
such steps, the usual minimum representing four series 
resistance steps and three parallel resistance steps in the 
control apparatus. The actual voltage variations are of no 
consequence, and the voltage per motor may with sufficient 
accuracy be considered as uniformly increased from zero 
to its final value. The motor voltage would then be repre- 
sented by a curve as OPQD in Fig. 50, the point P indicat- 
ing the time when the controller is full on. 

42. Motor Heating. — To ascertain whether a motor is 
suited for a proposed railway service, the conditions of that 
service must be investigated as already outlined. Barring 
commutation limitations, the motor must be large enough 
to dissipate the heat occasioned by the copper and iron 
losses without excessive temperature elevation. The cop- 
per loss over a typical run is equal to the product of the 
square of the effective motor current and the resistance 
of the motor windings. The iron loss depends upon the 
magnetic flux density of the iron and the armature speed. 
These in turn depend upon the current and upon the vol- 
tage impressed upon the motor terminals. If iron loss 



ENERGY CONSUMPTION. 1 19 

curves of the motor, for various current strengths, plotted 
in terms of motor impressed E.M.F. be available, a curve 
of iron loss for the time during which the power is on the 
motor can be constructed, since at each instant of time the 
motor current and voltage are known. Having subse- 
quently determined the average ordinate of such an iron 
loss curve over the entire run, that voltage may be found 
which, with the efTective motor current, will yield the 
same total iron loss. Thus, to reproduce the heating con- 
ditions of a proposed service in a shop test, it is only nec- 
essary to operate the motor for a sufficient time with the 
effective motor current value which gives the average cop- 
per loss at that voltage which gives with this current value 
the average iron loss. Such continuous operation should 
not result in a greater temperature rise than 75° C, start- 
ing cold. Should the calculations relating to a tentative 
equipment indicate a greater temperature elevation than 
this the motors must be discarded and a new set of cal- 
culations based upon larger units must be made. 

The equivalent voltage on 500 to 600 volt direct-current 
motors which yields the average iron loss does not vary 
widely and is generally somewhat less than 250 volts and 
rarely exceeds 350 volts even on interurban service with 
infrequent stops. Therefore, the continuous rating of a 
railway motor, which is stated in terms of the current which 
it can carry with a 65° C. temperature rise by thermometer 
at ^, f and full voltage, gives an accurate idea of its suit- 
ability for a proposed service. The nominal horsepower or 
kilowatt rating serves as an indication of the commutating 
limits and mechanical strength of the motor; it is the 
mechanical output at the car axle which causes a tempera- 
ture rise above the surrounding air (taken as 25° C), by 



120 TRACTION AND TRANSMISSION. 

thermometer, of not more than 90° C. at the commutator 
and 75° C. on any other accessible part after i hour con- 
tinuous run at rated voltage. 

43. Energy for Direct-current Propulsion. — The energy 
required for the propulsion of cars operating on direct 
current may be derived from the current and voltage curves 
of the motor equipment. The watts input to a car at any 
instant is equal to the line voltage times the instantaneous 
current per car. Thus a curve of power input can be plotted 
in terms of time. The area of this power curve would 
represent the electrical energy consumed during the run. 

Having already determined the average current per car 
in the foregoing numerical illustration over a particular 
run, namely 51 amperes, it is only necessary to multiply 
this value by the line voltage and by the total time of the 
run to obtain the energy consumption. Thus the electrical 
energy consumed is 

51 X 600 X 144= 4,410,000 watt-seconds, 
= 1,225 watt-hours, 

= 1.225 kilowatt-hours. 

In order to effect comparisons between different equip- 
ments as to economy of operation the energy consumed must 
be based upon some definite distance, such as a one-mile 
run. Energy consumption in kilowatt-hours per car-mile 
serves as a fair basis of comparison for cars weighing approx- 
imately the same but operating at different schedule speeds. 
When the car weights also differ considerably, the basis of 
comparison should be the energy in watt-hours per ton-mile. 

In the particular case of the 24.32 ton car making the 
0.8 mile run under consideration, the energy consumed may 
be expressed as 



ENERGY CONSUMPTION. 121 



-^ — ^ ~ 1-53 kilowatt-hours per car-mile, 
0.8 



or 



— ^ — = 6s watt-hours per ton-mile. 

0.8 X 24.32 ^ ^ 

44. Energy for Alternating-current Propulsion. — Be- 
cause of the varying power factor, the calculation of energy 
consumption of alternating-current railway equipments is 
not as simple as for the direct-current equipments so far 
discussed. The process of constructing current curves for 
the portion of the run after the period of constant accel- 
eration, that is, when full voltage is impressed upon the 
motors, is exactly the same as for direct-current equipments. 
The initial portion of the current curve, which refers to 
the current intake while the car is accelerated uniformly, 
is, however, difficult of exact determination in the case of 
alternating-current railway apparatus. The current taken 
by a single-phase motor increases somewhat during the 
period of constant acceleration, then decreases again before 
the expiration of this period, as shown by the curve in 
Fig. 51. In this figure are shown curves of speed, total 
motor current, volts on motor, and motor power factor, 
which were obtained by test with a 50-ton car, equipped 
with four 75-horsepower, single-phase motors, over a two- 
mile run. 

In the absence of such experimental data, the current 
taken by the motors of a proposed equipment for the 
period of uniform acceleration may be assumed constant 
at the value corresponding to the speed at which the rate 
of acceleration diminishes. On this assumption the current 
curve would have the form OABC, the portion BC being de- 
termined from the motor performance curves corresponding 



122 TRACTION AND TRANSMISSION. 

to the various speed values. The average motor current, or 
the effective current over the run, is then readily obtainable. 

Since the voltage impressed upon an alternating-current 
series motor at the first controller notch is usually about 
one-half of its final value, no great error will be introduced 
in the calculation of energy consumption by assuming that 
the motor voltage increases from the above-mentioned start- 
ing value uniformly to its maximum value. Thus, in Fig. 5 1 , 
the actual voltage curve HG may with sufficient exactness 
be replaced by the curve EFG. The average motor voltage 
may in this way be found without a knowledge of the 
experimental irregular curve. 

By making these two assumptions regarding the form of 
those portions of the current and voltage curves which 
correspond to the period of uniform acceleration of the car, 
the power taken by motors may be computed in kilovolt- 
amperes. The power factor of alternating-current railway 
motors on full voltage at different current inputs is em- 
bodied in the performance curves thereof. On the lower 
motor voltages during the period of initial acceleration the 
power factor is low. Its value during this time may be con- 
sidered as increasing uniformly to its value at full voltage, 
with the supposedly constant accelerating current, from a 
value at starting equal to about 40 % of its value at the 
instant when full voltage is applied to the motors. Thus, 
the actual power-factor curve of the motors shown as PMN 
in Fig. 51 may be represented by the curve LMN, This 
assumption makes possible the calculation of the power 
input to the motors in kilowatts. The total energy con- 
sumption of the equipment may be obtained by adding 
to this motor input the losses in the compensators, trans- 
formers, and car wiring. 



ENERGY CONSUMPTION. 



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124 TRACTION AND TRANSMISSION. 

45. Effect of Operating Conditions on Energy Consump- 
tion. — In order to determine the effect on the energy 
consumption of a railway equipment when the operating 
conditions are changed, such as altering the initial rate of 
acceleration, the length of run, the number and duration 
of stops, the gear ratio, the braking rate, and the line 
voltage, it is necessary to consider how the total energy 
taken from the trolley or third rail is expended. The energy 
supplied to a car or train during acceleration changes the 
momentum thereof, and the greater part of this energy ap- 
pears in the kinetic form, the remainder being expended 
in overcoming train resistance and in heating the starting 
rheostats and motor circuits. In bringing the car to rest 
subsequently the kinetic energy must be dissipated. Left to 
itself, the car would continue to move until all its energy 
of motion is lost in overcoming train resistance, and if, as 
is the usual case, the car is quickly brought to standstill 
after coasting for a time, the greater portion of the kinetic 
energy is consumed in heating the brake shoes and car 
wheels. Thus, the energy supplied to railway equipments 
is the sum of (a) the energy required to overcome the train 
resistance throughout the entire run, (b) the energy 
wasted in the starting rheostats, motors, and car wiring, 
and (c) the energy consumed in braking. 

A slight reduction in train resistance such as might be 
effected by the employment of ball or roller bearings in 
diminishing bearing friction, permits of a higher rate of 
acceleration with the same motor current. The greater 
the acceleration rate the more coasting is possible on a 
given run for the same schedule speed and the shorter is 
the time during which the motors receive power. A con- 
siderable saving of energy may result from the reduction 



ENERGY CONSUMPTION. 



125 



of train resistance to a minimum. With a given equipment 
the energy expended in overcoming train resistance is 
approximately constant for a given run. 

The energy lost in the starting resistances is proportional 
to the time that these devices carry current. The losses 
in the car wiring are usually small enough to be neglected 
in considerations of this kind. The motor iron losses and 
the loss in the gears are practically constant over the period 
during which the power is on. The copper loss in the motors 



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10 



0.5 1.0 1.5 2.0 

MILES PER HOUR PER SECOND 

Fig. 52. 



2.5 



3.0 



is proportional to the square of the current, and therefore 
the higher rates of acceleration with the accompanying 
larger currents result in a greater loss and consequent 
increase of heating in the motors. On the other hand, 
increased acceleration implies a shorter time during which 
the motors receive energy, and therefore tends to reduce 
heating. These two opposing conditions suggest that there 
is a definite rate of acceleration which will yield a minimum 
heating in a given case. 



126 



TRACTION AND TRANSMISSION. 



The energy consumed in braking depends upon the brak- 
ing rate and upon the speed of the car when the brakes 
are appHed. More coasting is permissible on a given run 
when high braking rates are employed, and the car speed 
at which braking begins is lower. Braking immediately 
after turning off the power and thus bringing the car to 
rest slowly results in inefficient operation. 

The curves of Fig. 5 2 show the motor current during the 
period of initial acceleration, the time of running on resist- 
ance, and the speed of the car at the instant of full- voltage 
application to the motors, in terms of the acceleration rates, 
for the 24.32-ton car already mentioned, which is equipped 
with four 50-horsepower, direct-current motors. These 
curves are plotted from the following data taken from the 
characteristic curves of the motors, Fig. 23. 



Acceleration 
rate. 


Total tractive 

effort per 

motor. 


Accelerating 
current. 


speed at full voltage 

with initial accel. 

current. 


Running time 
on resistance. 


.25 


222 


27.0 


31.8 


127.2 


.5 


374 


34.7 


25.5 


51.0 


.75 


526 


42.2 


22.0 


29.4 


I.O 


678 


49-5 


19.7 


19.7 


1.25 


830 


56.3 


18. 1 


14.5 


1-5 


982 


64.0 


16.9 


11.3 


1-75 


1134 


70.6 


16.0 


9.1 


2.0 


1286 


11-1 


15.2 


7.6 


2.25 


1438 


85-0 


145 


6.5 



(Train resistance taken as 70 pounds.) 

The curves verify the foregoing general statement that 
the greater the rate of acceleration the larger will be the 
current during uniform acceleration of the car but the 
shorter will be the time during which this current flows; 
and they show the dependence of these factors upon the 
rate of acceleration for this particular equipment. The 



ENERGY CONSUMPTION. 



127 



maximum schedule speed possible on any given run is a 
direct function of the rates of acceleration and braking. 

The maximum possible schedule speed increases with 
larger runs, provided all other conditions remain unaltered. 
Thus, in the case of the 24.32-ton car to which frequent 
reference is made, the relation between maximum schedule 
speed and the length of the run on level track, allowing for 



40 



,30 



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0.5 



1.0 



1.5 
MILES RUN 

Fig. 53. 



2.0 



2.5 



3.0 



20-second stops but no coasting, is shown in Fig. 53. This 
curve is based on data obtained from Fig. 31, on which a 
number of braking curves may be drawn corresponding to 
runs of various lengths. 

Proportionately less of the energy taken from the supply 
circuit is used to overcome the losses in other than train 
resistance for long runs than in short runs, and therefore 
the power consumption per mile is decreased by increasing 
the lengths of runs. This is also shown in Fig. 53 for the 
particular car under consideration; the curve of power con- 
sumption per car mile without coasting was computed in 



128 
50 

40 



TRACTION AND TRANSMISSION. 



■30 



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u 
o 



Q-20 



10 













^ 


^/ 


-^ 


6-CARTRAlNr4 MOTOR CARS.-154 .\>^y^ ^.^-^ 

TONS. AVG. BRAKING RATE1.75 vK^ .x'<f!l^^* 
MILES PER HR. PER SEC. STATION^V X^^ 1 
STOP-1 2 SECONDS. aP/ .'T^ 1 






















^11 








1 , 1 






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orjf 


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1 1 


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0.5 1.0 1.5 2.0 

RATE OF ACCELERATION IN MILES PER HOUR PER SECOND. 

Fig. 54. 



50 



40 



o 













^ 


ST^' 


r^TwT^ 


6 CAR TRAIN, 4 MOTOR CARS, 154 
TONS. ACCELERATION 1.33, 
MILES PER HOUR PER SECOND, 
STATION STOP 10 SECONDS. / 


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0.5 1.0 i.5 2.0 

RATE OF BRAKING IN MILES PER HOUR PER SECOND 

Fig. 55. 

connection with Fig. 49. The effect on schedule speed and 
on energy consumption of changes in the rates of accelera- 
tion and braking is not as conspicuous on long runs as 
on short ones. 



ENERGY CONSUMPTION. 



129 



The schedule speed of railway cars depends to a great 
extent upon the duration of the stops for the purpose of 
taking on or discharging passengers or freight. Obviously, 
the longer the period of standstill the lower will be the 
maximum schedule speed attainable by a given equipment. 

An increase in the time of coasting results in a reduction 
of the power consumption. The results of a series of tests 
on a 6-car train of the elevated railway in New York City 



50 



40 



1-30 

z 

hi 
O 
OC 
S2O 



10 



6-CAR TRAlNr4 MOTOR CARS,-154 TONS. 

AVQ. BRAKING RATE1.75 MILES PER HR. PER SEd. 

STATION STOP-14 SECONDS. 




2 3 4 5 6 

TIME IN SERIES POSITION, SECON DS. 

Fig. 56. 



made by H. S. Putnam are embodied in the curves of 
Figs. 54, 55 and 56, which show for a given schedule 
speed the influence on the percentage of coasting and per- 
centage saving in electrical energy, of acceleration and 
braking rates, and of running time in series position. 

The m.otor performance curves and the speed and power 
curves derived from them refer to a definite and constant 
trolley voltage. In practice this voltage has not the same 
value at different points on the roadway, owing to the drop 



I30 TRACTION AND TRANSMISSION. 

of potential along the trolleys, on third rail, and on feeders 
from the substations. The minimum voltage at the car 
should not be less than 350 volts for the usual 600- volt 
equipment. Consequently in selecting the car equipment 
for a proposed railway service due attention must be given 
to the voltage regulation on various parts of the road. 

Speed curves of cars operating on road sections on which 
the voltage is lower than normal must be based upon the 
average voltage existing at the definite locality. With 
series motors the speed at constant load varies almost 
directly with the impressed voltage, and hence the speed 
of the car at the instant full line voltage is applied to the 
motors is lower when the line voltage is below normal. 
Thus to maintain the same services under reduced voltage 
requires that the motor receive power for a longer time, 
and this frequently implies a greater power consumption. 
Sufficient trolley voltage all along the car route is impor- 
tant, particularly so on grades. 

46. Gear Ratio. — When a railway motor takes a cer- 
tain current at constant voltage a definite torque is devel- 
oped, and the corresponding tractive effort produced by 
the motor at the base of the car wheels depends entirely 
upon the gear ratio, that is, the ratio of the number of 
gear teeth to motor-pinion teeth. The resulting speed of 
the car for this motor current is inversely proportional to 
the tractive effort, and consequently the smaller the gear 
ratio the higher will be the speed of the car and the lower 
will be the tractive effort available for acceleration. There- 
fore, to maintain a specified initial rate of acceleration 
requires a larger current through the motors when geared 
for high car speed than when provided with a large gear 
ratio (i.e., low car speed). On the other hand, the time 



ENERGY CONSUMPTION. 



131 



that power is on the motors of a car when operating over 
a given run is longer with high gear ratios than with low 
ratios. The effect of change in gear ratio on the rate of 
acceleration with a definite accelerating current and on 
the magnitude of this current with a definite acceleration 
rate, is indicated respectively in the two following tables 
which refer to the 24.32-ton car equipped with four 50-horse- 
pow^er, 600-volt, direct-current motors whose characteris- 
tic curves are shown in Fig. 23 for a gear ratio of 17 to 69 
(or 4.06). 



Gear ratio. 


Rate of 
acceleration. 


1-5 
2-.0 

4.06 
5.0 


0.48 
0.68 
1.08 

1.50 

1.87 





Accelerating 


Gear ratio. 


current per 




motor. 


1-5 


142 


2.0 


IIO 


3-0 


80 


4.06 


64 


5-0 


55 



(Accelerating current 
= 64 amperes per motor.) 



(Acceleration rate = 1.5 miles 
per hour per second.) 



By constructing speed and power curves over a typical 
run for a given equipment when supplied with different 
gears, and subsequently plotting curves of power consump- 
tion and of effective heating current in terms of gear ratio, 
that gear ratio for the equipment can be determined 
which is conducive to a minimum expenditure of energy 
and least heating of the motors. In general, it develops 
that the most suitable gear ratio for motors of proper 
capacity for a specified service is that which will yield the 
lowest car speed consistent with the prescribed schedule 
speed, due allowance being made for delays. A gear ratio 
so chosen will result in a low energy consumption by the 
motors and a small temperature elevation. 



132 TRACTION AND TRANSMISSION. 

PROBLEMS. 

24. Upon the speed curve of Problems 17 and 18 plot the curve of current 
and power input per motor car. In determining the speed of the car at 
which the transition from the series to the parallel connection of the motors 
is made neglect the motor voltage drop. Compute the average current and 
power input per motor car over the time of the complete run. 

25. Calculate the energy consumption, in kilowatt-hours per train-mile 
and in watt-hours per ton-mile, of the train considered in Problems 17, 18, 
and 24. 

26. How much energy in kilowatt-hours is consumed by the equipment 
of the 20-ton car mentioned in Problems 14 and 15 over the run for which 
the service conditions are there specified? What is the equivalent heating 
current on this particular run? 

27. Determine from the curves of Fig. 51 the energy consumption in 
watt-hours per ton-mile of the 50-ton car equipped with four 75-horsepower, 
single-phase motors. Add 8 % of the power taken by the motors to allow 
for other losses in the car equipment. 

28. Plot curves of initial current, full voltage speed with initial accelerat- 
ing current, and time of running on reduced voltage, all in terms of the 
rate of acceleration, for a 100-ton New Haven electric locomotive equipped 
with four 250-horsepower, single-phase motors whose characteristic curves 
are shown in Fig. 25. Assume train resistance uniform at a value of 15 
pounds per ton. 

29. Construct a curve showing the maximum schedule speed possible, 
in terms of the duration of a stop, for the car whose typical speed curve on 
a level track is shown in Fig. 31. 

30. A motor car, weighing 43 tons, equipped with two 200-horsepower 
motors (gear ratio 20 : 63), whose characteristic curves are shown in Fig. 24, 
gains velocity at the rate of 2 miles per hour every second on a tangent level 
track. Assuming train resistance as 15 pounds per ton, plot a curve of 
the accelerating current required per motor when the equipment is pro- 
vided with different gear ratios, in terms of gear ratio. 



THE DISTRIBUTING SYSTEM. 133 



CHAPTER VII. 
THE DISTRIBUTING SYSTEM. 

47. Classification of Conductors. — It is common to 
divide the conductors of the distributing system into two 
parts, the ones which convey current from the station to 
the cars being termed positive and those which return it 
being tervntd negative. 

The positive conductors may be divided into three classes 
as follows: (i) bare contact conductors, such as trolley 
wires, third rails, and T conductors in slot systems, from 
which the current for propulsion is taken by means of 
collecting devices; (2) supplementary conductors, which are 
parallel to the contact conductors, are connected with them 
at frequent or infrequent intervals, and which are designed 
to increase or supplement their conductivity; and (3) feeders 
which extend from the station to a feeding point on the 
contact or supplementary conductors, and which supply 
current to them. 

The negative conductors may be similarly classified, 
although the bare conductor which receives current from 
the car is not usually termed a contact conductor. It 
usually consists of the connected track rails, although it 
may be a second trolley wire or T conductor in a slot 
system. Negative feeders and supplementary conductors 
are also common. 

The contact conductors are usually divided into successive 

sections each one of which is insulated from adjacent sec- 
10 



134 



TRACTION AND TRANSMISSION. 



tions. Their lengths vary from a few hundred feet to sev- 
eral miles. 

48. Contact Conductors. — To determine the drop as- 
sume a contact conductor BD^ Fig. 57, fed at B with / 




Fig. 57. 
amperes, 7i and I2 amperes being drained from it at dis- 
tances from B of h and h feet respectively. If the specific 
resistance of the conductor be p ohms per mil-foot and its 
cross section be A circular mils, then the drop from B to 
D is 

or 

g = -2- (/ J, + ij^) volts. 

Similarly in general, if any number, n^ of currents of dif- 
ferent magnitudes 7^ be drained off at different distances, L 
from B, the total drop from B to the most distant point of 
drainage may be expressed as 

'=j% ^^-im) = j ^0 i; (/J = ^ hi volts, (i) 

where U = yl ? is such a distance from B that if the 
total current / were carried that far the resultant drop 



THE DISTRIBUTING SYSTEM 13S 

would be e volts. The relations which exist between the 
currents, the distances, and /o are so similar to those which 
exist between the elementary and total masses of a body, 
the respective distances of the former from a plane, and the 
distance of the center of gravity from the plane, that the 
point which is h feet from B is termed the center of gravity 
of the combined drainage load. 

The total drop in a section of contact conductor is almost 
always assumed. Taken together with the drop in the 
negative part of the system it must not be so great as to 
hinder the proper starting and operation of the motors and 
the proper functioning of the lamps. The maximum drop 
in the negative conductors is usually made small with a 
view to meeting municipal ordinances or to preventing 
electrolytic corrosion. In England it is limited to seven 
volts. The total drop varies from 10 % to 50 % of the nor- 
mal voltage, the smaller value ruling in all alternating-cur- 
rent and in urban direct-current systems, while the larger 
is found in direct-current interurban systems. Knowing, 
therefore, the value of e, if the length of conductor and the 
distribution of the load be given, the proper cross section 
may be determined from (i) as 

A = -IqI circular mils. (2) 

e 

The minimum cross section of the contact conductor is 
dictated by mechanical considerations in the case of trolley 
wires, and by manufacturing standards in the case of third 
rails. The size of trolley wires is usually Nos. 000 or 0000 
B. &. S., although No. o has been used. With double- 
track roads and those single-track roads which employ 
twin trolley wires the sum of the cross sections of the two 
wires should be taken. If, therefore, the cross section, the 



136 



TRACTION AND TRANSMISSION. 



drop, and the load distribution be known, the limiting 
length of contact conductor which can be fed from a single 
feeding point may be determined by means of formula (i). 
The specific resistance of third rails varies with their 
chemical composition. Armstrong recommends the follow- 
ing limitations as to ingredients: 

Carbon not to exceed 0.12 per cent 

Manganese not to exceed 0.40 " " 

Sulphur not to exceed 0.05 " " 

Phosphorus not to exceed o.io " " 

Such compositions result in a resistivity of approximately 
14 microhms per centimeter cube at 20° C, a value which 
is seven and three-quarters that of commercial copper. The 
following table of rail resistances is based upon this value: 

RESISTANCE OF THIRD RAILS INCLUDING BONDS 



Rail weight in pounds 
per yard. 


Resistance in ohms 
per mile. 


40 


0.093 


50 
60 


0.074 
0.062 


70 
80 


0.053 
0.046 


90 
100 


0.042 
0.038 


IIO 


0.034 



Inasmuch as the current taken by a car varies with the 
time and location of the car and, in congested districts, is 
subject to further variations due to traffic conditions and 
the idiosyncrasies of the motorman, it is customary to 
assume a uniform drainage of /o amperes per foot from 
the contact conductor when treating urban or suburban 
problems where several cars are taking current at the same 
time from the same section. The value of Iq changes 



THE DISTRIBUTING SYSTEM. 



137 



during the day, and for calculating limiting conditions 
the rush-hour value should be taken. Its average value 
may be determined by multiplying the average current 
taken by each car in passing over the section by the num- 
ber of cars on the section at one time and dividing this 
product by the length of the section. The ratio of its 



o.v 


\ 




















2 5 


\ 
























\ 


















2.0 




\ 


\ 




















\ 


\, 














1.5 








\ 


\ 






















"^ 


-^ 








1.0 























10 20 30 

NUMBER OF CARS. 

Fig. 58. 



40 



50 



maximum to its average value may be determined by 
reference to Fig. 58, which is based upon experience. 

End Feeding. , Consider a section of length L feet, fed 
at one end as in Fig. 57, and let it be uniformly loaded. 
Since the current density /o = I/L, and the distance of the 
center of gravity of the aggregate load h = L/i^ the total 
drop over the section is 



whence 



6 = --/ = -7— /o volts, 
A 2 A 2 



L =t /i^feet. 
^ ph 



(3) 



(4) 



138 TRACTION AND TRANSMISSION. 

The total drop is therefore proportional to the square of 
the length of the section, and the maximum permissible 
length of section is to be obtained by use of equation (4). 

Center Feeding. If the section be fed at its middle point 
instead of at the end, the permissible length of contact 
conductor section is twice that indicated by equation (4). 
Such a system is schematically represented in Fig. 59, and 
is considered ideal from an operating viewpoint, for each 
section may be controlled by a circuit breaker at the station 
in the feeder supplying that section. This gives complete 



+ BUS 



BREAKERS^^^ 




Fig- 59- 

control of each and every section in case of overload, short 
circuit, accident, or repairment. It is the system most 
frequently used for urban roads. It may be desirable to 
connect the adjacent ends of the sections of the contact 
conductor through a section breaker which may be located 
on a near-by pole. When these circuit breakers are closed 
there results an equalization of the current distribution and 
the conductivity of the whole positive system becomes 
available. The remoteness of these breakers from the 
station, however, is objectionable as lacking accessibility. 
Watts Lost in Conductor, While the cross section of the 
contact conductor is usually prescribed by the maximum 
permissible drop or by mechanical considerations, cases 
may arise where a larger cross section will prove more 



THE DISTRIBUTING SYSTEM. 145 

and since nl = x, the distance in feet of any chosen point 
from the remote end, substituting the value of n 

fCx 
yx= \ ~ circular mils. (4) 

To determine the value of C, consider that the drop in an 
element of length dx at a distance x from the remote end is 

de = — = p/o \/— Vx dx; 

yx ^ C 

therefore the total voltage drop is 

e = p/o y ^ f x^dx = p/o y - - L^ volts. (5) 

Since the total entering current, /, is equal to the product 
of /o and the total length of the section, L, the value of 



n/' 



from equation (5) becomes 
1/ 



IC 2 pIVl . 



consequently 

y^ = Vx circular mils. (6) 

This equation shows that the curve which relates total 
cross section of supplementary and contact conductor with 
distance from the remote end is a parabola with its vertex 
at the remote end. Of course it is not practicable to con- 
struct a conductor with such a varying cross section, but 
it is common to reduce the cross section by steps as the 
remote end is approached. 

The connection of the supplementary to the contact con- 
ductor at many points involves considerable expense espe- 
cially when made through contact switches. It is therefore 



146 



TRACTION AND TRANSMISSION. 



common practice to employ a moderate number of connec- 
tions and to feed sections at each end and often from 
separate substations. In many instances this arrangement 
is used when the load is concentrated rather than uniformly 
distributed. In such cases the determination of the proper 




SUPP. COND. 



CONTACT 



CONDUCTOR 



SUBSTATION 



tkt 



I^ 



SUBSTATION 



TRACK 




Fig. 64. 

disposition of copper is involved and is best arrived at by 
trials based upon assumed distributions of copper and of 
load. 

Assume a system connected as in Fig. 64 which is elec- 
trically equivalent to the arrangement shown in Fig. 65, 





)' 



Fig. 65. 

where the resistances of the various branches and the 
voltage at the substations are known and the equivalent 
resistances R of the load and x of the rest of the conducting 
system, out and back from both substations and considered 
as connected in parallel, are to be found. The problem is 
solved by applying Kirchhoff's laws, which result in the 
following equations, where the resistances 



THE DISTRIBUTING SYSTEM. 



i 





A = a + b + c 






B=d+f+g 


ohms. 




C = b + d + h 




Ah 


-bis +IR = E 




Bh -dh -IR =-E 


-bh 


-dh +Ch = o 


h 


-h 


= I\ 



147 



(7) 



(8) 



Solving for R by means of determinants 
A o -b E 



R = 



B -d- E 
■b —d C o 

1 —I o / 



A -b (E-AI)/! 
B -d -Ell 
-{b-\-d) C b 



-b I 




A 


B 


-{b+d) 


-d -I 

/-I 




- b 


-d 

— I 


c 


C 




' I 

















ohms. (9) 



A o 

B 
-b -d 

1 —I 
Whence the voltage impressed upon the load is 

E{b+dY-E{A+B)C-{Ad-'+Bb''-ABC)I . . 

^^= ib + df -{A+ B)C ^°^''- ^'°^ 

The drop e between either substation and the load is 

e = xl = E- RIvolts, (11) 

where x is the equivalent resistance in ohms of the con- 
ducting system between the substations and the load. The 
drop between a substation and any point with a plurality 
of variously located loads is equal to the sum of the drops 
produced by each load. 

52. Graphic Time-table. — Since the reason for the 
employment of supplementary conductors is the preven- 
tion of an excessive drop of voltage between the substa- 
tions and the cars, the conductors must be of adequate 



148 TRACTION AND TRANSMISSION. 

cross section to cope with the worst condition likely to 
arise in the operation of the electric railway. As the 
voltage drop varies with the current and with the resist- 
ance, and the latter is proportional to the length of the 
conductors, the worst condition will be when a maximum 
total current is taken by cars at a maximum distance from 
both substations. To determine this condition use is 
made of graphic time-tables or train-sheets for the proposed 
service; such a curve is shown in Fig. 66. It consists of a 
set of intersecting curves, each one constituting the locus 
of the correlated time and place relations of a car or train. 
The ordinates may represent the hours of the day, while 
the abscissae represent distances from the road terminus 
in miles. The curves are usually considered as made up 
of straight-line elements. With equal scales for ordinates 
and abscissae the cotangent of the angle between a por- 
tion of the curve and a parallel to the axis of abscissae 
represents the corresponding speed in miles per hour. If 
the elements be straight the speed is constant, and in 
plotting these curves the average running speed is assumed 
to be maintained throughout. The perpendicular elements 
represent stops of durations proportional to the lengths of 
the elements. The ordinate of a point where two curves 
cross each other gives the time when the corresponding 
cars meet each other, while its abscissa determines the neces- 
sary location of a turnout, if the road have but a single 
track. For a specific ptoblem the time-table should have 
indicated upon it also the distribution of copper and the 
location of towns, villages, and substations. 

Confining the attention to a single section of the road, 
and assuming an average value of current taken by a car 
when running and another greater value when starting, the 



THE DISTRIBUTING SYSTEM. 



149 




3iNIl 



11 



150 TRACTION AND TRANSMISSION. 

magnitudes of the currents and the distances from the sub- 
stations of their points of drainage, corresponding to any 
chosen time, can be readily obtained. A comparison of the 
results for different times readily reveals the worst condi- 
tion likely to arise. With single-track interurban roads 
giving infrequent train service such condition is Hkely to 
occur when and where two trains pass each other. 

Having determined the worst condition, the adequacy of 
the assumed distribution of copper can be determined by 
the method outlined in the preceding section. The mini- 
mum voltage permissible at the car on 600- volt systems is 
300 volts, or with high-class service 350 volts. 

In the case of a supplementary conductor with numerous 
connections with a contact conductor which extends between 




I I I I I 



I I I I I 



L-l, 




y 



Fig. 67. 

two substations and is fed by both, the drop produced by 
a concentrated load is proportional to the current and to 
the distance from the nearer substation. Consider the 
conditions as represented in Fig. 67. If i? be the resistance 
in ohms per foot of combined conductor, the drop is 

e = Rhh = Rh{L - h) volts. (i) 

But / = /i -t- /2 amperes; (2) 

hence e = Rl(i — -zjh volts. (3) 

Therefore, for a given current /, the drop increases with 
increase of h from /i = o to /i = - . These equations also 



THE DISTRIBUTING SYSTEM. 151 

show that the portions of the current supplied to a car by 
the two substations vary inversely as their respective dis- 
tances from the car. 

53. Feeders. — Although supplementary conductors are 
often termed ^^ auxiliary feeders'' or simply ^^ feeders," the 
latter term is used in this text to represent conductors 
which extend from the station to a single feeding point 
and which carry the same current at the same time 
through every cross section. The cross section of a feeder 
is often determined from economical considerations and by 
the use of Kelvin's law as modified by Kapp: The most 
economical area is that for which the annual cost of energy 
wasted is equal to the annual interest on that portion of 
the capital outlay which can be considered proportional to 
the weight of metal used. 

Let / = maximum current in amperes carried by the 
feeder, 
L = length of feeder in feet, 
A = its cross-section in circular mils, 
h = effective annual hours of operation at maximum 

current, 
p = resistance of feeder in ohms per mil-foot, and 
w = weight of a mil-foot in pounds. 

Then the resistance of the feeder is -— ohms, and, if the 

A 

cost per kilowatt-hour delivered to the feeder be Cs dollars, 
the annual expense for energy lost in the feeder is 

= CspM^ dollars. (i) 

^ 1000 A 

At a cost of C2 dollars per pound of feeder conductor and 



152 TRACTION AND TRANSMISSION. 

at a rate for interest and depreciation of p^, the annual 
charge against capital outlay for feeder conductor is 

C/' = P2C2WLA dollars. (2) 

With overhead construction the cost of insulators and of 
installing the feeder will be independent of the cross-section 
for a specific case. Therefore the most economic cross- 
section is that which will make C/ + C/^ a minimum, in 
which case C/ = C/^ and the economic cross-section is 



= / \ / ^-^ — circular mils. (3) 

V 1000 i?2C2W 



1000 P2C2W 
Hence the maximum economic drop is 



The reciprocal of the radical in equation (3) may be termed 
the economic current density. Often the maintenance of 
a suitable operating voltage or the inevitable heating of a 
feeder precludes the use of the economic cross section. Long 
feeders may be fed from a special bus at the station at a 
potential somewhat in excess of the normal station voltage. 

In case the feeders are to be placed underground, an 
expression must be obtained for the annual expense charge- 
able against the cost or rental of conduit ducts in terms 
of the feeder cross-section. This expression must then be 
added to equations (i) and (2) before differentiating in order 
to obtain a minimum. 

Boosters, — In the case of feeding points remote from the 
station the cross section of feeders as prescribed by the 
permissible drop may be very large and may entail an almost 
prohibitive first cost. The cross section may be materi- 
ally reduced if a booster be inserted in the feeder circuit. 



THE DISTRIBUTING SYSTEM. 153 

Whether or not a booster should be used depends upon 
its cost and the expense of its operation and maintenance 
as compared with the saving resulting from the reduced 
feeder cross section. The determination of the advisabil- 
ity of its use and of its voltage may be made as follows, 
neglecting the losses in the booster: 
Let X = maximum voltage of booster, 

Cf = maximum total drop in boosted feeder, 
/ = maximum amperes in feeder, 
pi= interest, depreciation, etc., on cost of booster, 
/ and g = cost constants. 

Then 

Capacity of booster = ^K.W. 

1000 

Ix 

Cost of booster = f + g dollars. 

1000 

Hence the annual interest and depreciation on the booster is 

Ci = Pi{f + g-^^l dollars. 
\ 1000/ 

If h be the yearly effective hours of feeder operation and 
C3 be the cost in dollars of generating a K.W.-hour, the 
annual cost of energy lost in the feeder is 

C2 = ^^-^^^±^ hcs dollars. (5) 

1000 

If the length of the feeder be L feet, and its weight be 
w pounds per mil-foot, its cross section is 

A = — circular mils, (6) 

X + 6/ 

and its weight is 

W = —, — pounds. (7) 

x + e/ 



154 TRACTION AND TRANSMISSION. 

At a cost of Ci dollars per pound and a rate of interest, 
etc., of p2 per cent, the annual feeder expense is 

• C3 = '-^^^^^^^' dollars. (8) 

The total annual feeder and booster expense therefore is 

C = Ci + C2 + C3, 
or 

C = ^f/+g^+^>+^^^ + ^-2^^^^^^^ dollars. (9) 

\ 1000/ 1000 X + Cf 

In order that this expression may be a minimum its differ- 
ential coefficient with respect to x must equal zero, or 

dC ^ gl ^ Ihcs C2p2'WpIL^ 

— = pi H — 7 — , — r^ = o; 

ax 1000 1000 {x + e/y 

therefore 

. .2 _ CipiWpU 1000 

and 



x=Lv/-^5^^-^^ -..volts. (10) 
V pig + czh ^ ^ 

Since x must be a positive quantity, that value of L which 
makes it equal to zero is the minimum length of feeder with 
which the use of a booster is advisable. It should be noted 
that this minimum length increases as the yearly hours of 
boosted-feeder operation increase. Boosters are therefore 
to be especially recommended for intermittently operated 
feeders. If the average efficiency of the booster set be e, 
multiplication of the term c^h in (10) by (2 — e) will include 
the losses of the set. 

With the following values for the constants — those in 
brackets being suggestive of the order of magnitude — 
equation (10) may be simplified for use with copper feeders: 



THE DISTRIBUTING SYSTEM. 155 

p = 10.5. ^3= [0.006]. 

w = 0.00000303. pi= [o.io]. 

C2= [0.17]. / = [300]. 

p2= [0.06]. . g = [28]. 



X = 0.018 Lv/—- — , —€/. (11) 

V 2.8 + 0.006 ^ ^ ^ ^ 

For a total boosted-feeder drop of 50 volts and continuous 
operation oi h = 24. X 365 = 8760 hours, the minimum 
length of feeder to be boosted is found by making x = o. 
It is 

L = 20,650 feet. 

An infrequent operation would indicate a poorer load factor 
and accordingly higher cost per kilowatt-hour Cs. Assum- 
ing h = 1000 hours and Cs = o.oi the minimum length 
becomes 

L = 10,000 feet. 

54. Track Rails. — The size of track rails is determined 
by consideration of the mechanical requirements of the 
rolling stock, the schedule speed, and the character of 
ballast. The common sizes weigh from 60 to 100 pounds 
per yard of length. The specific resistance varies with 
the chemical constitution and, as carbon and manganese 
are usually present to the extent of about one-half per cent, 
amounts to about 20 microhms per cubic centimeter, 
while that for standard copper at 0° C. is 1.594. It is 
convenient to assume that for average temperatures it is 
ten times that of commercial copper. 

The usual length of a rail is 30 feet, although twice this 
length is sometimes used. In order satisfactorily to return 
the current to the station from the car, the rail sections 
must be conductively connected with each other by means 



156 



TRACTION AND TRANSMISSION. 



of bonds. These bonds are often made of copper, which 
has a much larger temperature coefl&cient of expansion 
than steel. As a consequence, it is not easy to maintain 
a good electrical contact between a copper bond terminal 
and the rail, under varying temperatures and the displace- 
ments caused by trafl&c. Many forms of these bonds have 
therefore been devised. The most satisfactory forms have 
their terminals either brazed to the rail or mechanically 
expanded in a hole in the web or flange of the rail. When 
heavy current-carrying capacity is desirable and the den- 
sity of traffic warrants the expense the rail sections may be 
welded to each other. 

It is desirable to use a pair of bonds for each joint, when 
they are of copper, to insure continuity of the circuit in 
case one bond should fail. With such bonding the resist- 
ance per mile of 30- foot rails may be assumed as 10 % larger 
than if the rail were continuous. 

For convenience in calculating the voltage drop in tracks 
the following values for the resistance of two track rails in 
parallel including that of 9-inch bonds of half the carrying 
capacity of the rail are given: 

RESISTANCE OF TRACK RAILS INCLUDING BONDS. 



Weight of rail, 


Resistance per mile, 


pounds per yard. 


ohms. 


40 


0.066 


50 


0.053 


60 


0.044 


70 


0.038 


80 


0.033 


90 


0.030 


100 


0.027 


IIO 


0.024 



THE DISTRIBUTING SYSTEM. 1 57 

55. Negative Track Feeders. — In those systems which 
make use of the earthed track rails for returning current 
from the car motors to the generating station, differences 
of potential exist between different points along the rails; 
as a consequence, the neighboring soil takes a part in the 
conduction of the return current owing to the presence in 
it of moisture, of dissolved substances, and of pipes or other 
metallic subsurface structures. At the points where the 
current leaves the last to enter the connection from the 
negative bus at the station, electrolytic corrosion occurs 
to an extent dependent upon the ampere-hours conducted. 
It is therefore desirable that this leakage current from the 
rails should be made as small as possible. Its magnitude 
is dependent upon that of the potential differences along 
the rails, and varies inversely as the resistance offered 
by the earth. It is not often that the engineer can alter 
the earth resistance, but he can materially vary the poten- 
tial distributions along the rails by using negative sup- 
plementary conductors or feeders, connected to the track 
at predetermined points, which serve as auxiliary return 
conductors. Owing to the large cross section offered to 
the current by the earth, its chief resistance, outside of 
that existing at the ground plate for the negative bus at 
the station, is that due to the layers of soil in the immediate 
vicinity of the rails, and this may be, and hereinafter is, 
considered as a transition resistance of a ohms per foot 
length of track (two or four rails) and varying inversely as 
the length. In the case of a track whose rails are connected 
to the ground and to the negative bus at the power house, 
if the excesses of potential, e, of the various points in the 
track above that of the negative bus be represented by the 
ordinates of the curve of Fig. 68, while the abscissae repre- 



iS8 



TRACTION AND TRANSMISSION. 



sent distances in feet from the power house, then the 
leakage current diej escaping at the point / to the soil 
from an elementary length, dl^ of track, is represented by 
the proportionality 

(i) 



alecc . 

a ' 



and the total leakage current is proportional to the area 



50 



o '^ 
< w 

Z liJ 
bJ > 

> < 

\- CD 

UJ LlI 

CD Z 

CO o 

> 





















^^ 














/ 


1 


(X 


y^^^^ 














^ 


y^ 




1 














/ 




r 






1 












y 


/ 








- 


1 


- 


-dl 








y 










I 


\ 











200 400 600 800 

DISTANCE FROM POWER HOUSE, h IN FEET. 

Fig. 68. 



1000 



included between the potential curve and the axis of 
abscissae, or 

leoz- I edL (2) 

a Jo 

In order to compare the relative merits for the reduction 
of leakage current of various proposed dispositions of the 
same amount of return copper, it is desirable that analyti- 
cal expressions be obtained for e in terms of the distances, 
/, from the power house for each proposed disposition. 
Substitution can then be made in (2) and that disposition 
which yields the minimum value of the integral may be 
adopted. 

As an illustration, consider a single generator supplying 



THE DISTRIBUTING SYSTEM. 159 

/ amperes to trolley feeders for a single-track road extend- 
ing L feet in only one direction from a station, the load 
being uniformly distributed along the line. Assume that the 
negative terminal of the generator is grounded at the station 
and that one negative supplementary conductor of uniform 
cross section, and bonded to the rails at short intervals, 
extends from the station to the end of the line. 
Let / = distance in feet of any point on the line from the 
station, 
i = current at this point in amperes, 
e = voltage of track at this point above negative 

terminal of generator, 
r = resistance in ohms per foot of return, including 

rails and negative supplementary conductor, 
p = ohms per mil-foot of copper, 
Ai= copper cross section in circular mils equivalent 

in conductivity to the track rails, 
Ac= cross section of negative supplementary conduc- 
tor in circular mils. 



i = I li --) amperes, (3) 



Then 



e = firdl = T-^ (l - -4) volts. (5) 

Jo Ai + AA 2L/ ^^^ 

The curve coordinating voltage to distance is therefore a 
parabola, and the area contained between it and the / axis, 
that is, the value of the integral in equation (2), is 



X 






i6o 



TRACTION AND TRANSMISSION. 



George I. Rhodes has compared various dispositions of 
return copper and concludes that a maximum reduction 
of leakage current can be obtained by the use of several 
insulated negative feeders of such cross section that the 
average potentials at their feeding points are maintained 




12 3 4 

NUMBER OF NEGATIVE FEEDERS 
Fig. 69. 

equal, the negative bus bar being insulated from the ground 
at the station. 

If, in addition, use be made of negative boosters in the 
feeders, the potentials at the feeding points can be main- 
tained uniform with that of the negative bus-bar even with 
widely fluctuating loads. 

The amount to which the original leakage current is 
reduced by various numbers of such negative feeders and 
boosters as a percentage of what would exist in the case 
of no feeders, is shown in Fig. 69. 



THE DISTRIBUTING SYSTEM. l6l 

If the contact-conductor sections be supplied by individ- 
ual feeders and the current of each be passed through the 
field exciting coil of the booster which is connected to the 
track feeder for the corresponding section, as indicated in 
Fig. 70, the potential of the track feeding points can be 
kept practically equal to that of the negative bus at the 
station. It should be noted that the track rails are insu- 
lated from the negative bus. This arrangement of connec- 



































r 


-r^ 


\GENERATOR 






(4 


-? 


1^ 


}\ K 






UJ 


U 


±\ 




N> 


-. 




p^f/p^ 


OOSTERS 














CM) 


Jll_ 










1 

































NEGATIVE TRACK FEEDERS 
Fig. 70. 

tions is the most effective one for minimizing electrolytic 
corrosion in those systems which return current through 
the grounded track rails. 

56. Electrolytic Surveys. — The determination as to 
whether and to w^hat extent track feeders shall be installed 
depends upon the conditions which result from the opera- 
tion of a road. These conditions are usually found by mak- 
ing an electrolytic survey and studying the results thereby 
attained. The difference of potential between the tracks 
and the various pipe systems is measured at many points 
throughout the roadway. Care must be taken that good 
terminal contacts be secured, for these differences seldom 
amount to more than a few volts. Upon a map, which 
clearly shows all the tracks, the potential differences are 
plotted as ordinates with respect to the track as abscissae, 
and a curve is drawn through their ends. Wherever the 



l62 TRACTION AND TRANSMISSION. 

track is positive with respect to the pipe the area included 
between the curve and the track is generally colored blue. 
In case it be negative the area is colored red, indicating 
that the potential conditions at such places are favorable 
to corrosion of the pipes. 

Another map is prepared from which the tracks are 
omitted but upon which the pipe system under investi- 
gation is indicated. The magnitude and direction of the 
currents flowing in the pipes at various points, especially in 
the red districts, are obtained and are indicated on this map 
by arrows of proportionate length and direction. Currents 
may be measured by the drop-of-potential method, using 
a low-reading millivoltmeter. The portion of the pipe over 
which the drop is to be obtained must be insulated from the 
earth and therefore excavations are generally necessary. 
A study of this map is likely to reveal the location of points 
where electrolytic corrosion is likely to take place. Thus, 
if at two points on an unbranched pipe currents be simul- 
taneously flowing towards each other, the conclusion is 
inevitable that they both leave the pipe at an intermediate 
point. Again, if a large current flow towards a point where 
a smaller one is flowing in the same direction, the excess 
of the former must leave the pipe at intermediate points. 

A relatively high potential difference between a track 
and pipe does not necessarily indicate that a large current 
is flowing between them, for such would not be the case 
if the resistance offered by the soil were large. It may be 
desirable to know whether the current be large or not, and 
this can be determined by the use of Haber's earth ampere- 
meter. It consists of a wooden frame in which is mounted 
a plate of glass with a copper plate on each side of it. The 
free surfaces of the latter are covered with a thin layer of 



THE DISTRIBUTING SYSTEM. 



163 



paste, made of copper sulphate and 20% sulphuric acid, 
and held in place by parchment. This frame is buried in 
the soil transverse to the supposed path of current flow. 
Leads from the copper plates are connected with a milli- 
amperemeter which will indicate the flow of current through 
the soil. The device is non-polarizable, and experience 
shows that its presence in the soil does not distort the 
current flow-lines. 

In order to make the current measurements it is neces- 
sary to know the resistance per unit length of the pipe. 
This may be obtained from the following table published 
by Prof. A. F. Ganz, based upon a specific resistance of 
0.00144 ohm per pound-foot of cast iron and 0.000181 ohm 
per pound-foot of wrought-iron pipe. ^ 



WEIGHT 


AND RESISTANCES 


OF CAST- 


AND WROUGHT-IRON PIPE. 




Standard cast iron. 


Standard wrought 


Extra heavy wrought 








iron. 


iron. 


Inside 














diameter 














of pipe, 
inches. 


Weight 

per foot 

without hub 

pounds. 


Resistance 

per foot, 

ohms. 


Weight 

per foot 

without hub 

pounds. 


Resistance 

per foot, 

ohms. 


Weight 

per foot 

without hub 

pounds. 


Resistance ' 
per foot, 
ohms. 


i 






.84 


.000215 


I .1 


.000164 


I 






1.7 


.000106 


2. 2 


.000082 


li 






2.7 


.000067 


3-6 


.0000502 


2 






3-6 


.0000502 


5- 


.0000362 


3 


II. 


.000131 


7-5 


.0000241 


10. 


.0000181 


4 


18. 


. 000080 


10.6 


.0000171 


15- 


.0000121 


6 


31. 


.0000465 


18.8 


. 00000963 


29. 


.00000623 


8 


42. 


.0000343 


28. 


. 00000647 


43. 


.00000421 


10 


55- 


.0000262 


40. 


. 0000045 2 


54. 


.00000335 


12 


70. 


.0000206 


49- 


.00000369 


65. 


.00000278 


16 


109. 


.0000132 










18 


130. 


.0000111 










20 


151- 


.00000955 










24 


205. 


.00000702 










30 


294. 


. 00000490 










36 


408. 


.00000353 










48 


604. 


.00000238 











164 TRACTION AND TRANSMISSION. 

57. Alternating-current Distribution. — The voltage drops 
which occur with alternating-current systems are dependent 
not only upon the resistances of the conductors but also 
upon their reactances and the phases of the components of 
current. An adequate general treatment of the subject 
is out of place in this text. The methods of determin- 
ing line reactances will be given in a later chapter. The 
flexibility and cheapness of transformers permit of their 
extensive use for the equalization of potentials, whereas 
excessive copper or boosters are essential in direct-current 
systems. 

The high permeability and the hysteresis characteristics 
of steel track and third rails involve large drops when they 
carry alternating currents. Skin resistance becomes an 
important factor and it has been estimated that at frequen- 
cies of 15 and 25 the current confines itself to a peripheral 
depth of but 4 and 3 millimeters respectively. Disregarding 
any drop due to flux set up outside the rail, its impedance, 
according to Armstrong, is 8 times the ohmic resistance at 
25 cycles and 6.2 times at 15 cycles. 

PROBLEMS 

31. Calculate the resistance at 20° Centigrade of a 30-foot length of track 
rail weighing 700 pounds. Take 7.7 as the specific gravity of steel rail. 

32. How far from the terminus of a road is the last feeding point to a 
No. 0000 copper contact conductor supplying o.oi ampere per foot, if the 
potential at the feeding point is maintained at 550 volts and the drop in 
the contact conductor must not exceed 20 per cent? 

;^^. The two cross-bonded contact conductors of the Manhattan Ele- 
vated Railroad consist of third rails weighing 100 lbs. per yard. They are 
fed at both ends from substations which maintain a constant potential of 
625 volts. If the distance between substations be one mile and the current 
drainage from both tracks at maximum load be 0.3 ampere per foot, what 
is the maximum percentage drop in the contact conductors? 

34. Determine the economic cross-s^ctioij of a copper feeder to carry 



THE DISTRIBUTING SYSTEM. 165 

350 amperes for 2500 effective hours per year. Assume the cost of a kilo- 
watt-hour as one cent, the cost of a pound of copper 18 cents, and the rate 
of interest and depreciation as 6 per cent. 

35. If the feeder of problem 34 be supplied with current at 550 volts, 
what is the greatest length which may be used without producing a drop 
exceeding ten per cent? 

36. Plot a curve, based upon the constants given in § 53, which shows 
the dependence of equivalent hours of operation upon the minimum feeder 
length for economic installation of a booster assuming an average booster 
efi&ciency of 85 per cent. 

i 



12 



1 66 TRACTION AND TRANSMISSION. 



CHAPTER VIII. 
SUBSTATIONS. 

58. Types of Substations. — A substation is a station 
which contains devices which serve to alter the voltage or 
character of the current received from the transmission line 
and thereafter deliver it to the distributing system. Sub- 
stations are of three types, depending upon the character 
of the received and delivered currents as to whether they 
are direct or alternating. 

59. Direct Currents Received and Delivered. — With 
the Thury system, which is employed to some extent in 
Europe but which is not looked upon with favor by Amer- 
ican engineers, direct current is generated at the power 
house, transmitted and received at the substation and 
direct current is sent out from the substation. A typical 
example of this system is the plant which transmits power 
from Moutiers in Savoy to Lyons for the operation of the 
street railways in the latter city. Sixteen water-turbine- 
driven direct-current generators, consisting of four sets of 
four each, are connected in series with each other and can, 
at full load, generate 3500 volts each or 56,000 volts in all. 
They supply a constant current of 75 amperes to the line, 
and their voltage is varied with the load by means of 
electrically operated regulators connected in series with the 
line. The sets may be operated singly or together accord- 
ing to the load requirements, a single movement of a 
controller handle on a simple switchboard serving to cut in 



SUBSTATIONS. 1 6/ 

or out a set. The transmission line is no miles long, con- 
sists of two copper wires 0.354 inch in diameter, and 
entails a constant loss of 535 kilowatts. It has been found 
necessary to keep the line connected to the earth through 
high resistances and to provide numerous lightning arresters. 
At the substation the received current is used to operate 
motors each of 540 horsepower capacity. The speed of 
the motors is maintained constant by centrifugal regula- 
tors which shift the brushes when the load changes. These 
regulators are criticized as being an inherent defect of the 
system, for they are complicated and frequently require 
adjustment and repairs. Each motor is used to drive a 
600-volt direct-current generator which is connected with 
the distributing system. Special precautions are taken 
to insulate the motors from each other, from the earth, 
and from the generators which they drive. Tests have 
shown that the power output of the substation is 0.705 
that of the intake of the turbines which drive the generators 
at the power house. As a precaution against breakdown 
of the line or power station, the substation is amplified by 
an auxiliary transformer station in which direct-current 
motors are direct connected to 10,000-volt three-phase 
generators, the latter being adapted for connection with 
the hnes of another operating company. These sets are 
reversible and by means of them energy may be supplied 
to or received from the other system. The power stations 
and the substations in this direct-current system cost more 
than those which use alternating currents for transmission. 
The cost of the transmission line is less and the maximum 
voltage, as limited by the appearance of corona, § 72, is 
greater. The system is lacking in that flexibility which 
characterizes the use of transformers. 



i68 



TRACTION AND TRANSMISSION. 



60. Alternating Currents Received and Delivered. — In 

those systems which employ induction motors on the cars 
or locomotives, three-phase currents are generated at the 
power station, and, if the length of the transmission line 
requires more than an impressed voltage of 12,000 — the 
upper voltage limit of generators — at least three single- 
phase step-up transformers or one three-phase transformer 
must be used. At the substation three step-down trans- 
formers must be located, and usually a fourth one is in- 
stalled as a spare unit. Such substations are designed to 

1.00 



>0.99 
o 

■z. 
u 

o 
u_ 
lijO.98 



0.97 







i 






































































^ 


js^Xj^ 


t£5-— 












^ 


^-x^ 











































250 500 

CAPACITY IN KILOWATTS. 

Fig. 71. 



750 



operate without an attendant and therefore the transformers 
are self-cooling and both the primary and secondary circuits 
are supplied with automatic oil switches adjusted to open 
on short circuits but not on overloads. Fig. 71 shows the 
full-load efficiencies of a Hne of 25-cycle. 11. 000- volt air- 
blast transformers of capacities from 100 K.W. to 750 
K.W. The buildings are of fireproof construction, and 
permanently installed ammeters and voltmeters facilitate 
the location of possible faults on the system. 

In those systems which employ single-phase commutator 



SUBSTATIONS. 1 69 

motors, if the transmission line be single phase and be 
long, and consequently the voltage be high, but one step- 
up and one step-down transformer are necessar}'. Since, 
however, it is cheaper to use a three-phase transmission 
line it is ad\'isable to use a three-phase generator and three 
step-up transformers at the power station and two step- 
down transformers at the substation, the latter being con- 
nected according to Scott's method for transformation 
from three-phase to two-phase with connections as shown 
in Fig. 12. Furthermore, the cost per kilowatt of three- 
phase generators is but about three-quarters that of single- 
phase generators, because in the former a single magnetic 
circuit is used in common by all phases. 

Experience has shown that it is practicable to use 
alternating-current pressures as high as 20,000 volts on 
overhead contact conductors. In such cases stationary- 
substations may be dispensed with, and voltage reduction, 
suitable to the requirements of the motors, can be attained 
by the use of transformers located on the cars or locomo- 
tives. In some respects this arrangement is ideal, each 
motor ha\'ing a substation and carn.'ing it with it. There 
are no substations on the electrically equipped portion of 
the X. Y. X. H. vSc H. R.R.. 11.000 volts being generated 
and impressed directh* upon the contact conductors of the 
system. Each motor, however, is pro\-ided with a trans- 
forming de\'ice. The locomotives used in the Berlin- 
Zossen tests were equipped with pohphase motors wound 
for an impressed pressure of 10,000 volts taken direct from 
the contact conductors without the inter\'ention of voltage 
transforming de\'ices. 

61. Alternating Currents Received and Direct Cur- 
rents Delivered. — Substations which convert altematino^ 



I/O TRACTION AND TRANSMISSION. 

current into direct current are the type most frequently 
used. By means of transformers the voltage of the currents 
received from the transmission line is stepped down and the 
secondary currents are supplied to converters or motor- 
generators which deliver direct currents to the distribut- 
ing system. The motor element of the motor generators 
may be either a synchronous motor or an induction motor. 
The proper selection of the conversion apparatus involves 
a number of considerations. 

Floor Space, — In all cases it is customary to install 
three single-phase transformers or one three-phase trans- 
former for each converter. Since both induction and syn- 
chronous motors are wound for an impressed E.M.F, up 
to 12,000 volts, step-down transformers can usually be 
dispensed with. Even then the floor space occupied by 
converters and transformers is less than that required for 
equivalent motor-generators. Wilson and Lydall give the 
following values for units of about 750 K.W. capacity: 

Converters and transformers, 0.21 sq. ft. per K.W. 
Induction motor-generators, 0.31 sq. ft. per K.W. 

The possible separate location of converters and trans- 
formers, for instance the placing of transformers on a 
gallery, gives a flexibility of arrangement of apparatus not 
possessed by motor-generators. With urban substations 
and expensive real estate the occupied floor space becomes 
an important factor. 

Efficiency. — The efficiency of synchronous converters is 
greater than that of motor-generators. Even if to the 
losses of the converters be added the losses in transformers 
and regulating devices, which are not involved in the use of 
motor-generators, the efficiency of the combined converter 



SUBSTATIONS. 



171 



installation excels. W. R. C. Corson gives the average 
operating efficiencies from this point of view as follows: 

Synchronous converters 91% 

Synchronous motor-generators 85% 

Induction motor-generators 84% 

Figs. 72 and 73 contain curves showing the operating 
characteristics of a shunt-wound, 25-cycle, 600-K.W.conver- 



2500 




25 50 75 100 125 150 175 200 

PER CENT, OF FULL LOAD CURRENT 

FROM COMMUTATOR 

Fig. 72. 

ter, and of a 50-cycle, 230-K.W. induction motor-generator 
respectively. 

Regulation. — Since the ratio between the commutator 
and slip-ring voltages of a converter is practically constant, 
irrespective of the field excitation, except in the case of 
spHt-pole converters, it is customary to insert a reactance 
coil in the circuit between the low-tension terminal of a 



1/2 



TRACTION AND TRANSMISSION. 



transformer and the converter slip ring which it supplies 
with current, and to provide the converter with a series 
magnetizing coil which is traversed by the direct current 
from the commutator before it enters the feeders of the 
distribution circuit. The field excitation is thereby caused 
to increase with load, and the alternating current which 
enters the slip rings is therefore made to lead the impressed 
voltage. The passage of the leading current through the 
reactance coil establishes such phase relation that the vector 



o 

z 0. 

UJ 

5 

Lu 0. 

Ll. 

UJ 



o 

d: 

LlI 

5 0.1 
o 











^Y 




















^sjf5 


fp^ 


"0^ 






~~~~ 


~~^ 


==^ 


::^ 




A 


^ 


c^ 


















/ 


:/ 






















i^ 






































1 











100 200 300 

LOAD IN KILOWATTS. 

Fig. 73. 



400 



sum of the transformer and reactance voltages is greater 
than the former and therefore the slip-ring voltage is raised 
with load. The converter with such an arrangement is 
said to be compounded^ and may maintain a constant direct- 
current voltage under wide variations of load. It is 
usual to provide for each phase a reactance coil of a 
combined kilovolt-ampere capacity equal to 15 % of the 
rated kilowatt capacity of the corresponding converter. 
Fig. 74 shows a General Electric Company air-blast reac- 
tance set and starting switches for a looo-K.W., six-phase 
converter. The operating characteristics of the 600-K.W. 



SUBSTATIONS. 



173 



converter previously mentioned, with added series ampere- 
turns at full load amounting to 64 % of the shunt ampere- 
turns, are shown in Fig. 75. With proper adjustments of 
the series and shunt field coils it is possible to make the 
converter take a lagging current on light loads and a leading 
current on heavy loads. It therefore increases the power 




Fig. 74- 

factor of the transmission circuit on heavy loads. This 
method of regulation, however, fails to give satisfactory 
results when the line resistance drop exceeds 10 % of the 
impressed line voltage or even less; and yet on large trans- 
mission systems and with long transmission lines it is desir- 
able and often economical to have a drop greater than this. 
With motor-generators, however, the direct-current volt- 
age can be as easily and satisfactorily regulated as with 



174 



TRACTION AND TRANSMISSION. 



plain generators, and the regulation is in nowise dependent 
upon the drop in the transmission line. Furthermore, by 
the use of series coils on a synchronous motor field the 
motor-generator set may be adapted for power factor correc- 
tion to the same extent as with converters. 

Cost, — The cost of converters per se is less than that of 
motor-generators of the same capacity. Compound con- 
I00| \ \ — ^^n ~L \ \ ^ ^--03 



< ■ 



o 

CL 

a 

'2L 
< 

O 

z 

LJ 
O 
L. 
U. 
LJ 





y-'^ 


r- 


-—- 


-=. 


.^ 





} 


1 
1 


/ 1 












r 




Or/ 

o/ 
o/ 










f/ 




Ujj 


2j 








4 




f 










fr 




r 












w 


¥ 




y 








# 


.^^ 




^^1 


,;/ 


'/^>- 




j 


cp- 




J 






i '"^ 








.-^' 
.■^y 


t:^ 










K 

% 




y 









2250 



2000 c 

: 

1750^ 

L 
C 

1500. 
> 

1250 ^ 

c 

1000 = 
c 

L 

750 £ 
c 

500 < 
250 



25 50 75 100 125 150 175 200 

PER CENT, OF FULL LOAD CURRENT 

FROM COMMUTATOR 

Fig. 75. 



the 



verters cost more than shunt converters because of 
lower flux density in the iron. 

To make a proper comparison of the costs of the two 
types of installation one should consider the whole system 
and compare the total cost of converters, regulating devices, 
transformers, switch gear, ventilation apparatus, and trans- 
mission cables with that of equivalent motor-generators, 
switch gear, and cables. Parshall and Hobart make such a 



SUBSTATIONS. 



175 



comparison for a plant supplying three substations each 
having a rated output of 1800 K.W., the most remote being 
6 miles from the power house. The results are given in 
the following table. 

RELATIVE COSTS OF CONVERSION INSTALLATIONS 



High-tension cables 

Converters (6-900 K.W.) 

Motor generators 

Transformers and ventilating sets (21-300 

K.W.) 

Substation switchboards and gear 

Total 



Converters. 



$80,000 
67,500 



31^500 
27,000 



$206,000 



Motor 
generators. 



$55»ooo 
118,000 



15,000 



$191,000 



The smaller cable expenditure with motor-generators 
results from their ability to operate satisfactorily with a 
greater line drop than is allowable with converters. Whether 
the interest on the 7 % less outlay with motor-generators 
would offset the increased operating cost resulting from 
the smaller efficiency of the motor-generators would require 
a careful study of the substation load diagrams. The pre- 
ceding table is based upon the following costs per rated 
kilowatt: 

Converters $12.50 

Transformers and ventilation apparatus 5.00 

Converter switch apparatus 5.00 

Motor-generators 21.90 

Motor-generator switch apparatus s-SS 

The data concerning the converter equipment relate to 
an existing substation. 

62. Location of Substations. — There are certain points 
on the roadway of a traction system which may be con- 
sidered as natural points for the location of a substation. 



1/6 TRACTION AND TRANSMISSION. 

These are the centroids of load in urban networks, the 
power house when it is located on the line, and the middle 
or a point near the remote ends of the terminal sections 
of the lines. It is also often desirable to have the substation 
located at a passenger station, thus making it possible for 
the ticket agent to serve as a substation attendant. 

If it be assumed that there is a uniform drainage of cur- 
rent throughout the length of the road and that the con- 
tact conductor has numerous connections with the supple- 
mentary conductor, the composite conductor, of uniform 
cross section, extending from one substation to each adja- 
cent substation, then the economic distance between sub- 
stations can be determined by mathematical treatment. 




±1 



I COMPOSITE CONTACT CONDUCTOR } 

j^ . L .{ 

Fig. 75. 

Furthermore, if the profile of the road be such that along 
certain portions the drainage of current is greater than 
along the rest of the line, each portion by itself can be 
treated mathematically. 

Assume a road of length L feet to be supplied with cur- 
rent from n substations, equally spaced from each other 
by a distance X = L/n feet, and arranged as in Fig. 76, 
where the substations are represented by S, 

The annual mean effective current per foot of contact 
conductor can be determined from a study of the train 
diagrams and from the instantaneous currents per car. 
The maximum drop, which will occur at a point midway 



SUBSTATIONS. I J 7 

between substations and at the terminals of the Hne, is 
Kmited to such a value as will permit satisfactory operation 
of the motors and lighting of the lamps, is known, and 
must he used as a check on the economic drop about to be 
determined. See problem No. 37. 

For a fixed distance between substations, the economic 
cross section. A, for the composite contact conductor is 
such that the annual charge for interest and depreciation 
on its cost is equal to the annual charge for the energy lost 
in it. To prove this, consider that the former charge is 
dependent on the weight of the conductor, that is its cross 
section, and may be placed equal to KiA, and the latter on 
the resistance, which may be placed equal to K2/A , where 
Ki and K2 are constants. The sum of these two charges, 
X, must be a minimum, hence the differential of x, with 
respect to A, must equal zero. Therefore 

dx '^ K2 

and 

KiA = K2/A dollars. (i) 

If now, with a conductor of constant cross section, the 
distance between the substations be increased, which is 
equivalent to reducing the number of substations for a 
road of given length, the resistance and weight of the con- 
ductor between stations will be increased proportionately. 
The interest charge will likewise increase, while the energy 
charge will increase to a greater extent, because the current 
entering the section of conductor from the substation has 
also been increased. Therefore K2/A is, in this case, larger 
than KiA, and to maintain the equality of equation (i) 
the value of A must be increased. 



178 TRACTION AND TRANSMISSION. 

The increase of distance between substations, or reduc- 
tion in their number, furthermore affects the charges for 
interest, maintenance, and operation of all the substations. 
The wages for fewer attendants and the costs and losses 
per kilowatt of the larger units installed are thereby de- 
creased. The economic cross section of contact conductor 
and economic distance between substations, therefore, in- 
volves a minimum annual charge for wages, for interest on 
total cost of copper and equipment, and for cost of total 
energy lost in copper and equipment. Expressions for 
each of these items of annual charge must be found in 
terms of the distance, X, between substations, and the 
differential coefficient of their sum, with respect to X, must 
be equated to zero in order to determine the economic 
separation of substations. 

It will be assumed that the annual charges against the 
transmission line, the energy lost in the track, and the cost 
of substation buildings are not affected by changes in X. 
The last two charges can be introduced without difficulty, 
if desired. The first charge materially alters with X only 
in the case of very short lines and very heavy traffic. 

Wages. — For a given type of substation, length of line 
and density of traffic, the necessary number of attendants 
in each substation and their average wages will not vary 
with the size of the units, so far as these sizes are dependent 
upon X. For all substations, however, they will vary 
directly with the number of substations, n = L/X, and if 
there be n' attendants per substation, receiving on an 
average w^ dollars per year, the total annual charge for 
attendants 

Cy, = nn'w' = [n'w'L] - • (2) 

X 



SUBSTATIONS. 179 

With transformer substations there are no attendants and 
therefore C^ becomes^ in this case, zero. 

Charges against Contact Conductor, — Consider that part 
of the contact conductor of cross section A circular mils 
which is fed from one substation. Under the assumption 
of a uniform drainage of /o mean effective amperes per foot, 
the watts lost in each half of the conductor, or X/2 feet, are, 
according to equation (5), § 48, pl^^?/!^ A, There being 
8760 hours in a year, at a cost of c^ dollars per kilowatt- 
hour delivered from the substation, the annual charge for 
the energy lost in X feet of the conductor is 

C; = 8760 _p£3^ j,^, ^^jj^^^_ ^^ 

1000 12 A 

If the cost of conductor be C2 dollars per pound and w be 
the weight of a mil-foot in pounds, at an interest rate of po 
the annual capital charge against the contact conductor is 
C/' = p2C2wA\ dollars. (4) 

Since C/ must equal C/^ when the cross section A is most 
economical, equations (3) and (4) may be equated and 
solved for A as follows: 



A = 0.855 /oX\/ ^ ^ circular mils. (5) 

V P2C2W 

Substituting the value of A in (4), multiplying by 2 so 
as to include CJ and by L/\ = n to cover the whole length 
of line, the total annual charge against contact conductor is 

A A 

or Cc= [1.71 L/o Vpw/?2^2^3jX dollars. (6) 

Annual Charge against Substations, — If the total max- 
imum output of all substations be P kilowatts and if the 



l8o TRACTION AND TRANSMISSION. 

overload coefficient or ratio of maximum output to rated 
installed capacity be 5, then the rated capacity of the appa- 
ratus installed in each substation is P/bn K.W. The over- 
load coefficient is determined from a study of the nature 
of the load diagram for each substation and from the over- 
load guarantees as to the apparatus. In determining the 
number of units to be installed in each substation the fol- 
lowing points must be considered : 

{a) It is desirable and good practice to have the same 
sized units throughout the system whenever possible. 

{b) There are limits as to the maximum size of units 
to be found among manufacturers' standard lines. 

{c) The daily load curve is often of such a character 
that one unit and several units can be operated for pro- 
tracted intervals at nearly maximum efficiency. 

{d) The maintenance of the continuity of service requires 
that either a spare unit be installed in each substation or 
that there should be a portable substation which can be 
placed on a siding as needs may require. 

{e) The peak of the load may be taken by a storage 
battery installed in each substation. 

(/) Provision must be made for increased output with 
growth of traffic. 

Fig. 77 shows the load curve on No. 2 substation of the 
Manhattan Division of the Interborough Rapid Transit 
Company on a particular day. This substation was then 
equipped with six 1500-K.W. converters each having effi- 
ciencies of 93.5, 95.75, and 96.0 per cent at half, full, and 
five-quarters load respectively. They were supplied with 
alternating current from eighteen 550-K.W. transformers, 
three for each converter, each having efficiencies of 97, 97.75, 
and 97.7 per cent respectively at the corresponding loads. 



SUBSTATIONS. 



i8r 





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cr 

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13 



l82 



TRACTION AND TRANSMISSION. 



Assuming that the overload capacities are as recommended 
in the Standardization Rules of the A.I.E.E., that is, that 
they can each carry an overload of 25 % for two hours and 
50 % for one-half hour, the load diagram shows the prob- 
able operating conditions of these units on this day to be 
as in the accompanying table, the numbers in the third 
column indicating the equivalent number of hours that a 
unit must be operated at full load in order that its losses 
may be the same hypothetically as they are in fact. To 
determine the equivalent hours, if the efficiency at any load 
be €, let the expression (i — e) be termed the deficiency at 
that load; then the equivalent hours are equal to the pro- 
duct of the number of hours at any load by the ratio of that 
load times its deficiency to full load times its deficiency. 



CONDITIONS 


OF OPERATION OF UNITS 


Unit. 


Hours per day. 


Equivalent 
hours per day. 


No. I 


24 


21.4 


No. 2 


155 


150 


No. 3 


8.8 


7-5 


No. 4 


2.0 


2 .0 


No. 5 








No. 6 









Total daily equivalent unit, hours, 45.9 

The equivalent annual hours of operation of all units 
in this substation at full load are therefore 



^ = 365^^^ = 2792 hours. 

The load on this substation was about 20 % greater in 
winter than as shown in Fig. 77, due partly to the current 
required for car heaters. Instantaneous fluctuations of 



SUBSTATIONS. 



183 



current above and below those shown in the figure amounted 
in some cases to 40 %. In calculating losses in a proposed 
substation a mean effective load diagram should be used. 

To obtain an expression for the annual charge for energy 
lost in the substation in terms of X, it is necessary to plot 
deficiency curves in terms of the rated capacity of units. 
There should be say three curves, for half, full, and three- 
halves load respectively. The points on these curves can 



.GC 

.03 

.07 

O.06 



^.04 
Q 













^ 








.iAi 


































h^ 


Md 










-.^_^ 










.,..._____^ 




ztu 


l^^ 


y'-LlA-a-'^ 








^ 


— 








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3 




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^ 


5ac 




















^^0. 


^/2 


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^000^2^^ 


— . 




' 


^ 






__. 






























































CONVERTER -TRANSFORMER UNITS. 







































































500 



1000 



CAPACITY, Pq, in kilowatts. 
Fig. 78. 



1500 



2000 



be obtained readily from manufacturers' efficiency curves of 
units for say three rated capacities, as 500, 1000, and 1500 
kilowatts. Three such curves for combined transformer- 
reactance-converter units, at unity power factor, are shown 
in Fig. 78. The full-load curve is practically straight over 
the portion covered by the capacities entering into the 
problem, and the deficiency, 9, may be expressed analyti- 
cally as 

^=fs-gzPo. (7) 

where Pq is the rated capacity in kilowatts. 



i84 



TRACTION AND TRANSMISSION. 



The following values are suggestive of the order of magni- 
tude of the constants /s and gs for conversion at 25 cycles 
from 11,000 volts to 600 volts: 



DEFICIENCY CONSTANTS 



Units. 


K.W. 


A. 


g.v 


Transformer-reactance-converter 
Transformer-reactance-converter 
Transformers 


500 to 2000 
200 to 500 
100 to 750 


0.072 
0.087 
0.024 


0.000012 
. 000060 
0.000014 







The converters of larger capacity listed in the table are 
wound six-phase, while those of smaller capacity are three- 
phase. If there be u units of capacity Pq kilowatts in- 
stalled in each substation, including spare units, and h be 
the equivalent annual hours of operation of all units at full 
load, then, since Pq = P/dnu, the annual loss of energy in all 
substations is 



Pofiu^h = 



Pfsh P%h 
8 b-fiu 



kilowatt-hours. 



(8) 



Since n = L/\ if the cost per kilowatt-hour of energy 
delivered to the substation be c^ dollars, the annual charge 
against the substations for energy lost in them is 



c: = 



'Pcz'm 
. b _ 


— 





X dollars. 



(9) 



The cost of one unit of capacity i^o kilowatts can be 
expressed analytically as Jz + gs'i^o dollars, where fz and 
gz are constants determined by the manufacturer. The 
cost of all units to be installed in all substations is therefore 
nu{jz + Pgz /8nu), and, if /?3 be the annual rate covering 
interest, depreciation, and obsolescence, the annual charge 
against cost of substation equipments is 



SUBSTATIONS. 



I8S 



CJ' = PznuW + Ppzgz'/b, 
or, since n = L/\, 

C:' = [^] + [ZM'^] i dollars. (lo) 

The following values are suggestive of the order of magni- 
tude of the constants /s' and gs. 

COST CONSTANTS 



Units. 


K.W. 


U: 


gs'' 


Transformer-reactance-converter. . . 
Transformer-reactance-converter. . . 
Transformers 


500 to 2000 
200 to 500 
250 to 750 


3200 

2000 
240 


9-4 
II. 
2.66 



The total annual charge against the substation equip- 
ments is equal to the sum of CJ and CJ^ as given in equa- 
tions (9) and (10) J or is 

The Economic Spacing of Substations. — The economic 
value of X is such that the total annual charges or the sum 
of Cy,, Cc, and Cs, as given in equations (2), (6), and (11), 
shall be a minimum. To avoid needless repetition of the 
letters entering into the bracketed coefficients of these 
equations^ these coefficients may be represented as follows: 

and their sum as 

To determine the minimum value of X, the differential 



l86 TRACTION AND TRANSMISSION. 

coefficient of C, with respect to X, must be placed equal to 
zero, or 

Solving, 



^=\/§^^,ieet, (12) 



K. + K^ 
and substituting the values of the coefficients, 



X = / nWL + PsjVuL^ ^^^^ ^^^^ 

The economic cross section, A^ for the composite contact 
conductor can now be obtained by inserting the value of X 
in equation (5). 

63. Numerical Illustration. — For the purpose of more 
clearly understanding the influence of the factors entering 
into the economic spacing of substations, assume a road 
200,000 feet long with converter substations that are to 
be cared for by two attendants, each receiving $720 per 
annum, and each on duty 12 hours each day, every station 
to be equipped with two converter units of equal size. The 
cost and deficiency constants will be those applying to 
units under 500 K.W. capacity. Let the following be the 
values of the characteristic constants: 

P/b = 2500 K.W., b = 1.25, 

/o = 0.00875 ampere per foot, p2 = 0.06, 

h = 5000 hours, ps = o.io, 

p = 10 ohms, C2 = 0.18 dollar, 

w = 0.00000303 pound, Cs = C3 = O'Oi* 



t\ 



SUBSTATIONS. 



187 



Then 



X = ([2 X 720 X 200,000 +0.10 X 2000 X 2 X 200,000] 

-^ 1. 71 X 0.00875 X 200,000 X 

Vio X 0.00000303 X 0.06 X 0.18 X o.oi — 

. . ^ _ 0.01 X 0.00006 X sooo' 
(2500)2 X — 

200,000 X 2 

= ([288,000,000 + 80,000,000] 



=v/, 



-^ [2990 V0.OOOOOOOO327 — 0.0469]) 2 



368,000,000 
0.169 — 0.0469 



= 54,800 feet = 10.4 miles. 



Thus, the economic separation of converter substations on 
this 37.8-mile electric railway is 10.4 miles; consequently 

30 



J 29 



00 o 



J CO 28 

^027 

1-- 



26 



V 




\ 


\l 




\>^ l^'^ 







20 



35 50 65 

SPACING IN THOUSANDS OF FEET 
Fig. 79. 



80 



4 substations will be required, each equipped with two con- 
version apparatus units of 300 K.W. rated capacity. That 
10.4 miles is the economic distance between substations is 
proved by computing the various cost items for the railway 
which depend upon this distance for different values of X, 
as in the following table, and as shown in Fig. 79. 



i88 



TRACTION AND TRANSMISSION. 



Cost items. 


Substation spacings in feet. 


30,000 


50,000 


54,800 


60,000 


80,000 


Wages, Cio 

Copper, Cc 

Equipment, Cs. • . ■ 


$ 9,600 

5.075 

14,885 


$ 5,760 

8,460 

12,890 


$ 5.260 

9,280 

12,515 


$ 4,800 
10,130 
12,145 


$ 3.600 
13.570 
10,875 


Total 


$29,560 


$27,110 


$27,055 


$27,075 


$28,045 



Automatic substations, which dispense with operators 
and avoid their wages, are now in successful operation. 
At a predetermined low line voltage a contact-making volt- 
meter actuates a drum controller which, by the use of con- 
tactors, places a converter in service. When its output 
falls to a fixed value a current relay shuts it down. Pro- 
tective devices do the same in case of short circuits, defect 
in any device, or failure of supply voltage. 

64. Auxiliary Storage Batteries. — If a storage battery 
in series with a compound- wound booster^ be connected 
between the positive outgoing and negative incoming feeders 
of a substation, the two may be so adjusted as to impress 
a constant voltage upon these feeders. As a result, a slight 
decrease of converter voltage under abnormal load allows 
the battery to discharge into the distributing system, and 
also a slight increase of converter voltage under subnormal 
load will cause the battery to receive a charging current 
from the converter. The use of a battery, therefore, re- 
lieves the substation units, the transmission line, and the 
power station apparatus of violent instantaneous fluctu- 
ations of load. If the battery be of sufficient capacity, 

^ For a discussion on the connections and operation of boosters and stor- 
age batteries see Chapter VIII, Dynamo Electric Machinery, Vol. I, by 
Sheldon and Hausmann. 



SUBSTATIONS. 1 89 

it may also serve to carry the peak loads, of not too long 
duration, which are common on interurban systems. If, 
again, the battery be of very large capacity, it may serve 
to carry the characteristic peak loads of an urban system 
and may serve to supply power to the whole system in 
case of accident in the power station or on the transmission 
line. The use of a battery, therefore, may enable one 
to install smaller units in substations and in generating 
stations and to operate them under better load factors and 
therefore at greater efficiencies. It also enables one to 
design the transmission line for average instead of maximum 
load conditions. The saving in investment for station equip- 
ments and line must however be balanced against the cost 
of batteries and boosters, and the decreased energy losses 
must be balanced against the energy losses attendant upon 
the use of the battery. Furthermore, the cost of extra 
attendance entailed by the use of batteries must be consid- 
ered. The proper capacity of such a battery is so closely 
dependent upon the characteristics of the substation load 
diagram that the advisability of its installation can be 
determined only from the study of the specific case. 

65. Arrangement of Apparatus. — The arrangement of 
apparatus in a substation is governed to some extent by 
the cha-racter of the equipment and the size and shape of 
the available site. It is desirable to have all apparatus 
on one floor; but, if the equipment be large, the switch 
gear should be placed on a gallery so that the attendant 
may command a view of the whole station. In urban 
districts, where real estate is expensive, the transformers, 
high-tension switches, and lightning arresters are often 
placed on a second floor. Storage batteries when used 
in substations are usually located on another floor or 



1 90 



TRACTION AND TRANSMISSION. 




|.xp^\^\^v^^^^ 




^V^^\\^^^^^^\<xV^^^^^^ 



SUBSTATIONS. 



191 



in a separate building adjacent to the main substation. 
The path of energy from the transmission Hne to the dis- 
tributing feeders should be as short and direct as possible. 
This leads to the following arrangement across the station 



^^ .3ii<RE?^^ 




Fig. 81. 



from the transmission line: high-tension entrance devices, 
lightning arresters and switch gear, transformers, reactances, 
converters, low-tension switch gear, and outgoing feeders. 
Fig. 80 is a sectional view of a substation of the Milwaukee 
Electric Railway and Light Company. This substation 
has a rated capacity of 1200 K.W. for conversion from 
66,000 volts alternating current to 1200 volts direct current. 



192 



TRACTION AND TRANSMISSION. 




Fig. 82. 




Fig. 83, 



SUBSTATIONS. 



193 



Fig. 81 gives a view of the low- tension end of one of the 
substations of this road, the reactances surmounted by 
starting panels being shown as located in front of their 
respective converters. Fig. 82 shows the method of tapping 




the transmission line on the substation roof, and shows the 
high-tension roof bushings for insulating the supply wires 
at their points of entrance to the substation. Figs. 83 
and 84 respectively show the electrolytic lightning arresters 
and the high-tension oil switches and the methods em- 
ployed in their installation. 

66. Portable Substations. — On most electric roads 
there are certain sections of the line on which abnormally 



194 TRACTION AND TRANSMISSION. 

heavy traffic must be handled at infrequent intervals or 
only during a certain portion of the year, as for instance 
near fairgrounds, parks, or summer resorts. To meet such 
a condition and to guard against interruption of service 
due to accident to a unit in any substation, it is much 
cheaper to make use of portable substations than to in- 
stall permanent spare units. These substations consist 
of specially arranged cars containing complete substation 
equipments, of the converter or motor-generator type, 
with accessories. The standards as to track gauge, height 
of tunnels, and strengths of bridges limit their character- 
istics to 500 K.W., 60,000 volts, and 150,000 pounds weight. 
The external appearance of such a portable substation is 
shown in Fig. 85. The arrangement of apparatus is shown 
in the plan and elevation of Fig. 86, and Fig. 87 is a diagram 
of the circuit connections. The positive feeder cable is 
carried to a terminal block on the outside of the car near 
the roof, for convenient connection to the trolley wire or 
feeder. The incoming high-tension lines may be connected 
directly to the transmission line; but, if frequent or con- 
tinued use of the portable substation in one locality is 
necessary, disconnecting switches should be mounted on 
the nearest pole to facilitate disconnecting the oil switch 
without having to cut off power from the transmission 
line. 

The use of such portable stations insures continuity of 
supply with minimum investment in permanent substations, 
saves large investment in copper and equipment on lines 
infrequently loaded, provides additional capacity at any 
point where there may be a temporary abnormally heavy 
traffic, and may furnish power for extensions during the 
period of construction. 



SUBSTATIONS. 



195 




J 

'■■i 







196 



TRACTION AND TRANSMISSION. 





i 



T^ 



rfl^ 





SUBSTATIONS. 
PROBLEMS. 



197 



S7. Derive an expression for the economic spacing of substations, the 
cross section of the composite contact conductor being prescribed by a 
mean effective drop of e volts at a point midway between substations. 

Suggestion. — Obtain an expression for A by using (3) of § 48, insert it in 
(3) and (4) of § 62, which then add, multiply by L/X and use in place of 
(6) of § 62 for the economic determination. 



^i^:^k'/?^^^r T^T^/^sms^m //he 






ifFesi/dimce^ 





*-A/nm^t^r: 



fI/5^ 



Me^rat//?f 



% 



\(//TMrra/?jr^/'mr^ 



V^/t/n^tfr 




% rased 

-B/oiv^r 



7d£fa^7//zer\ \7d7r(/c/f^ 

Fig. 87. 

38. Assuming the same wages as in the illustration of § 63, what influence 
would a change of the hours of duty to eight hours a day have upon the 
spacing? 

39. If the road in this illustration were to be operated with single-phase 
currents and each substation were to be equipped with two single-phase 
transformers, what would be the economic spacing? Use cost and deficiency 
constants given in § 62 and neglect reactance drop in the distributing system. 

40. If all the equipment in all the substations of the road specified in 
14 



198 TRACTION AND TRANSMISSION. 

§ 63 were to be operated for 8760 hours per year at rated capacity, what 
would be the value of h and what would be the economic spacing, assuming 
1 200 as the generator voltage. 

41. If the costs of switch gear and lightning arresters for each substation 
on the road specified in § 62, were to be $2500 and $3750 for 250 K.W. and 
500 K.W. capacities respectively, what change would thereby be entailed 
in the cost constants /s' and gz and how would this change affect the value of 
the economic spacing? 



TRANSMISSION LINES. 199 



CHAPTER IX. 

TRANSMISSION LINES. 

67. Location of the Transmission Line. — In those in- 
stallations which employ steam or internal-combustion^ 
prime movers in the power station, it is desirable to locate 
the latter with reference to the substations which it suppHes 
with energy, so that a minimum weight of conductor mate- 
rial shall be required and the drop of voltage to each shall 
be the same. This location is termed the center of distribu- 
tion. Consider two substations X feet apart and distant Xi 
and X2 feet respectively from an intermediate power station. 
Assume the substations to be supplied, over a three-phase 
line, with h and I2 annual mean effective amperes per wire 
respectively. If the specific resistance of the conductor 
material be p ohms per circular mil-foot, and the respective 
cross sections be ^1 and A 2 circular mils, then the 
power lost in transmitting energy to the substations is 
Pi = 3 pXi/iV^i and P2 = 3 9X2^1 1 A<i watts respectively. 
If the weight of a circular mil-foot be w pounds, the total 
weight of conductors is 

W = '^w (Xi^i + X2^2) pounds. 

Substituting the values of Ax, A2, and X2 = X — Xi, 

TT^ W/l^ . (X^-2XXl+Xl^)/2^ , ,. 

W = gpw ( —- h ^^ p — j pounds. (i) 



200 TRACTION AND TRANSMISSION. 

For a minimum weight of conductor material, the differ- 
ential of W with respect to Xi must equal zero. Hence 

or '^ = '^' (2) 



Pi P: 



2 



If the drop to both substations be the same, PJIi = P2//2, 
and 

Xi/i = X2/2, (3) 

wherein Xi and X2 now represent the respective distances of 
the substations from the center of distribution. For any 
number of substations located at various points along a 
continuous roadway the distance of the center of distribu- 
tion from any point is Xo = 2X//2/, each length being 
measured along the path taken by the transmission Hne. 

The location of the power station at the center of dis- 
tribution is subject to other considerations, such as the 
cost of real estate, future growth, facilities for the receipt of 
fuel and supplies and the removal of ashes, and the avail- 
ability of water for condensing purposes. 

In the case of hydraulic installations, the location of 
the power station is dependent on the hydraulic conditions, 
and may be quite distant from the substations. The trans- 
mission line should extend from the central station to the 
nearest substation or to the one nearest the center of 
distribution, that route being selected which results in the 
lowest annual cost. 

Private rights of way for the transmission line are to be 
preferred to public highways and generally result in final 
economy in operation. Rights of way along steam rail- 
roads are undesirable because of insulation troubles likely 



TRANSMISSION LINES. 201 

to result from coal smoke. It is not practical to make the 
right of way so wide as to prevent a pole or tower from 
falling on the abutting property, but the right to trim trees 
on both sides should be secured. A width of from 50 feet to 
100 feet is ample. The cost of right of way is from 25 to 
50 per cent of the total cost of the transmission line. 

68. Number of Phases. — The proper basis for deter- 
mining the number of phases to be employed is the com- 
parison of the weights of conductor material necessary to 
transmit the same power, P kilowatts, over the same dis- 
tance, S feet, with the same loss, P^ watts, and the same 
maximum voltage, E kilovolts, between any two conductors. 
In a system using n wires each of cross section A circular 
mils and carrying / amperes the loss is 

P^=^^atts. ' (i) 

A 

npSP 
Therefore A = circular mils. (2) 

The total weight of the conductors is therefore 

W = nwAS = —z^n^P pounds; (3) 

that is, the weight is proportional to the square of the 
product of the number of wires by the current flowing in 
each wire. The following table is based upon the current 
per wire in amperes for transmitting, at unit power factor, 
one kilowatt with a loss of one watt per foot of line at one 
effective kilovolt between wires of greatest potential differ- 
ence. With direct currents the equivalent voltage is V2 
kilovolts. For the three-wire quarter-phase system, where 
the center conductor carries V2 times the current in the 
outer conductors, it is assumed that the cross sections of 



202 



TRACTION AND TRANSMISSION. 



the conductors will be so chosen that the loss per foot is 
the same, P' Iz-, ii^ ^^^h conductor. The maximum voltage 
between any two conductors is assumed the same in all 
cases because its value determines the capacity and cost of 
each insulator. The center wire of the three-wire quarter- 
phase system, however, does not need to be so well insu- 
lated as the outside wires, and to this extent the above 
comparison is unfair to this system. Considering, however, 
that the conductor expense considerably exceeds the in- 
sulator expense in most cases, this system does not need to 
be considered in comparison with the three-phase system, 
which, as shown in the table, is superior to all systems using 
alternating currents. 

RELATIVE WEIGHTS OF CONDUCTORS. 



System. 

Two wires: 

Direct current . . . . 

Single-phase 

Three wires: 

Three-phase 

Quarter-phase: 
Right-hand wire 

Center wire 

Left-hand wire . . 

Four wires: 

Quarter-phase. . . . 




TRANSMISSION LINES. 



203 



69. Frequency. — The Standardization Rules of the 
A. I. E. E. give 25 and 60 as standard frequencies. For 
transmission hnes supplying converting substations one or 
the other should be used. The weights and costs of 60-cycle 



50 


























40 


\ 
























30 




\ 






















?o 




^ 





























Uy 


^ 




~~-— -i 


5 CY'l 


JLES 










to 













1000 2000 

CAPACITY IN KILOWATTS. 
Fig. 88. 



3000 



transformers are less than those for 25 cycles and the oper- 
ating efficiencies of the former are greater than those of 
the latter. The differences are not very great, as will be 
seen from the curves in Figs. 88 and 89, which refer to 
1.00 

.99 



98 



o 

^.97 



.96 



.95 





































6( 


► CYC 


LES 












^ 


^^^ 


^ 


— ' 


2S 


> CYC 


LES 










y 


y 


^ 




















/ 

























1000 2000 

CAPACITY IN KILOWATTS. 

Fig. 89. 



3000 



33,000-volt, plain steel, air-blast transformers. Induction 
motors for higher frequencies are also cheaper, but operate 
at lower power factors. At the lower frequency it is less 
difficult to operate generators and other synchronous appa- 



204 TRACTION AND TRANSMISSION. 

ratus in parallel, because the unavoidable variations in speed 
are smaller in proportion to the angular velocity. The 
charging current of the line and the inductive drop are less 
with low frequencies, and may give a better regulation. 
For lines of moderate length it might prove desirable to 
use 60 cycles, but the general tendency is to use 25 cycles. 
For lines of great length, however, it is usually undesir- 
able to use 60 cycles for the following reasons. In all large 
systems odd harmonic frequencies of voltage and current, of 
which the third and fifth may predominate, are likely to be 
present and be superposed upon the fundamental frequency. 
Electromotive-force harmonics may be due to armature 
reaction, to pulsation of inductance, to the distribution of 
armature windings, or to non-uniform distribution of mag- 
netic flux in the air gaps of the generators. Current har- 
monics may result from similar causes associated with the 
structures forming the receiving apparatus. Every trans- 
mission line, because of its inductance and capacity, has a 
resonant frequency. The magnetic field of the former and 
the electric field of the latter serve for the storage of energy 
in kinetic and potential forms respectively. Such capacities 
for the storage of the two forms of energy are characteristic 
of every medium for wave propagation, and their magni- 
tudes determine the velocity of the propagation. As will 
be shown later, the velocity with which an impressed differ- 
ence of potential travels away from a generator along a line 
of usual construction is but slightly less than the velocity 
of light, that is, in the neighborhood of 186,000 miles per 
second. Now a transmission line with both ends open or 
both ends short-circuited has a resonant frequency which 
corresponds to a wave length equal to twice the length of 
the line, as is the case with an organ pipe open at both ends. 



TRANSMISSION LINES. 205 

On the other hand its length is but a quarter wave length 
when one end is open and the other short-circuited, as is 
the case with a closed organ pipe. In the latter condition 
a line 155 miles long would correspond to a wave length of 
4X 155 = 620 miles and the corresponding resonant fre- 
quency would be 186,000 miles per second divided by 620 
miles, or 300 per second, which is the frequency of the fifth 
harmonic, when the fundamental is 60 cycles per second. 
The use of 60 cycles on a line of such length is therefore 
likely to result in resonant oscillations of current and electro- 
motive force which may prove disastrous. 

For the operation of single-phase railroads a frequency of 
less than twenty-five permits of a marked reduction in the 
size of a motor for a given output ; and yet almost all such 
roads have adopted 25 cycles. The New York, New Haven 
& Hartford Railroad is an instance. The Midi Railway 
of France, among others, has adopted 15 cycles. A deter- 
mination of the most suitable frequency for such installations 
is desirable, involves extensive knowledge as to costs and 
peculiarities in operation, and must be considered as to its 
bearing on the general question of the standardization of 
practice. 

70. Economic Voltage. — The economic voltage between 
the wires of a transmission line depends upon the amount 
of power and the distance over which it is to be transmitted 
as well as upon the various cost factors of equipment and 
energy. To understand the method for its determination 
and to avoid complexity, assume a single three-phase line 
of equivalent length S feet supplying at a maximum P kilo- 
watts, divided equally among n substations, each of which 
contains two j:onverter units of rated capacity P/2 n kilo- 
watts. Assume further that the rated capacity of each of 



2o6 TRACTION AND TRANSMISSION. 

the three single-phase step-up transformers at the power 
station is P/3 kilowatts. 

Conductor Expense. — If the yearly mean eflfective power 
factor be cos and the voltage between wires be E kilovolts, 
the full load current per wire will be 

£ = __^ amperes. 

V 3 E cos (i> 

If the resistance of a mil-foot of conductor be p ohms, the 
resistance of each wire will be pS/ A ohms, and if the 
equivalent effective yearly hours of operation on full load 
current be h^ the annual loss of energy in all three wires 
will be 

3 RPh phSP^ , ., 

~ = ^-^o :r^ kilowatt hours. 

1000 1000 A E"^ cos^ 4) 

If the mean annual cost of delivering a kilowatt hour to the 
middle of the line be c^ dollars, the annual expense for 
energy lost in the line conductors will be 

C/ = '^f^r .. dollars. (x) 

loooylE^cos^ (t> 

If w be the weight of a mil-foot in pounds, C2 be the cost 
per pound, and p2 be the rate of interest and depreciation 
on the cost of conductors, the annual charge on the capital 
outlay for all three conductors is 

C/^ = 3 P2C2WSA dollars. (2) 

Since equations (i) and (2) must be equal to each other for 
a minimum annual cost, they may be equated and solved 
for A, giving 



A = — \/ — circular mils. (3) 

E cos 6 V ^000 i?9C9W 



TRANSMISSION LINES. 207 

Substituting this value of ^ in (2) and multiplying by 2 
so as to include (i), the total annual charge against the 
conductors will be 



(4) 



C, = C/+ C/^ = [ o.iogG PS V,3p/i^,,,ze;]i dollars, 
L cos jE 

and representing the bracketed expression by Kc, 

C, = K,/E dollars. (s) 

Pole and Insulator Expense. — There are several stand- 
ard forms of construction of towers or poles. Many rigid 
steel towers have been installed and recently flexible steel 
structures costing materially less than those of the rigid 
t}^e have been used with success. The determination of 
the type to be employed can best be made in connection 
with a specific problem, which determination will also give 
the economic distance, X' feet, between poles. With poles 
of the flexible type the cost, Cp, does not materially vary 
with the voltage between the line wires. Furthermore, 
if insulators of the suspension type be employed, the 
cost of each one per kilovolt, Ci, is practically constant. 
Since the number of poles to be used on a line of real length 
S^ equals S^ /W and the number of insulators is three times 
this, if the annual interest and depreciation on these items 
be pp and pi respectively, the annual pole and insulator 
expense is 

Cp = [p,c,S'/\'] + [3 PiCiS'/\'] E dollars, (6) 

and, representing the bracketed expressions by Kp and KJ 
respectively, 

Cp = Kp + KJE dollars. (7) 

Pin-type insulators cost more per kilovolt as the oper- 
ating voltage increases. It is assumed by some that the 
cost thereof increases as the cube of the voltage. 



2o8 TRACTION AND TRANSMISSION. 

Transformer Expense. The costs of transformers depend 
not only upon their rated capacity but also upon the volt- 
age at the high-tension terminals. The insulation expense 
increases with voltage. For the same capacity and voltage 
water-cooled transformers are cheaper than air-cooled ones. 
Power-station facilities are generally such as to permit the 
use of water-cooled step-up transformers, while air-blast 
transformers are common in substations. A study of the 
prices for transformers shows that the cost of each, c^, can 
be expressed by the following formula, where E represents 
the high-tension kilovoltage, Pi the rated capacity in kilo- 
watts, and K and K^ are constants: 

ct = {KE + K') VPi dollars. (8) 

This formula applied to transformers where Pi varies from 
500 to 4000 and E from 22 to 66, gives results within the 
variations between the quotations from different manufac- 
turing companies. It is approximately true also for higher 
voltages. In a particular problem with many substations 
it would be wise to make use of two sets of values for 
the constants applying respectively to the power and sub- 
station transformers. 

The number of transformers in the power station is 
three; each of capacity P/3 kilowatts. There are 6n in 
the n substations; each of capacity P/6n kilowatts. If 
pt be the rate of interest and depreciation on this apparatus, 
the annual expense for transformers in dollars is 



C, = 2>p, {KE + K') VP/3 + 6 p,n {KE + K') Vp/6 n, 
which by combining and transposing becomes 

c,=[3/.,rvp7^(i+v^)]+[3/.,i^Vp7^(i+v^)l£; 

(9) 



TRANSMISSION LINES. 209 

and, if Kt and K/ represent the bracketed expressions, the 
annual transformer expense may be represented as 

Ct = Kt + K;E dollars. (10) 

Auxiliary Expense, — The costs of aluminum lightning 
arresters, choke coils, and oil switches increase with the 
voltage of the circuits with which they are to be connected. 
The first mentioned increase more rapidly than the voltage, 
the second nearly directly, and the last less rapidly. If 
their combined costs for different voltages be determined, 
it will be found that the cost per three-phase unit may be 
expressed, with sufficient accuracy, as a linear function of 
the voltage. Considering a unit to consist of a four-tank 
arrester, three choke coils, and a triple-pole oil switch, and 
one unit to be installed in each substation and in the power 
station, if c^ be the cost per unit per kilovolt and pa be the 
rate of interest and depreciation, the annual expense charge- 
able to these auxiliaries will be 

Ca = \paCa{n-\-\)\E, (ll) 

and representing the bracketed expression by iT^, 

Ca = i^a^ dollars. (12) 

Solution, The economic voltage is now determined by 
adding the expressions for the annual expenses for con- 
ductors, poles, insulators, transformers, and auxiliaries, 
differentiating the sum with respect to £, equating to zero 
and then solving for E as follows : 

C={K, + K,) + KJE + {KJ + K:+ Ka)E dollars, (13) 
^ = -KJE' + {KJ + K: + Ka) = o. 



2IO TRACTION AND TRANSMISSION. 

Therefore the economic voltage between wires is 



Substituting the values of the constants from equations 
(4), (6), (9), and (11), 

_ / (0.1096 PS /cos (/)) Vcsphp2C2W 

" V 3 PiCiSy\'+3 ptK VP/^ (i + VTn) + p^Ca (n + i) 

kilovolts, (15) 

and the economic cross section of the conductors is found 
by inserting this value in equation (3). 

In the above derivation the total transformer capacity 
at the power station has been assumed equal to that in all 
substations. In existing plants the latter exceeds the 
former by from 40 per cent to 60 per cent. This is feasible 
when the load peaks of the different substations are not 
simultaneous. The ratio of the maximum load supplied at 
one time to all substations to the sum of the maximum loads 
on each substation is termed the diversity factor. Further- 
more, it has been assumed that the power factor at maxi- 
mum load is unity. This can be realized as resulting from 
the phase of the currents taken by converters at maximum 
load when the voltage regulation is that produced by 
reactances. The converters then tend to correct the power 
factor of the line. The energy given to the line at the 
power station must, however, exceed that which is deliv- 
ered to the substations by the amount which is lost in the 
transmission line. 

Generally a transmission line extends from the power 
station to one of several substations, then divides, and con- 
tinues to the other substations. The currents in the branch 



TRANSMISSION LINES. 211 

conductors are less than in the conductors of the main 
line and the cross section is accordingly reduced. The eco- 
nomic cross section of a conductor of a branch, of length Sb 
feet between substations, is determined by ecpuation (3) 
and the total annual charge against the conductors by 
equation (4) . If the mean annual effective power factor on 
the branch be the same as on the main line, then the main 
line may be considered as having added to it a length Se 
such that the annual conductor expense for the branch is 
included in that for the main line. Remembering that 
/ = P/ V3 E cos (/), and equating two expressions like equa- 
tion (4), applied to lengths Sb and Se and to currents Ib and 
/ respectively, 

IbSb — ISej 

whence Se = SbIb/I- (16) 

If the distance from the power station to the first sub- 
station be So feet, then the equivalent length to be used in 
calculating the annual expense of conductors is 

S = So + ^Se, (17) 

the last term including the extension of length due to all 
branches. 

In calculating the annual expense against insulators and 
poles, however, the real length of the complete Hne must be 
taken. 

71. Numerical Illustration. — Assume a single three- 
phase 25-cycle line having an equivalent length oi S = 
350,000 feet and a real length of S^ = 450,000 feet, trans- 
mitting, at maximum rated load, P = 3000 kilowatts 
divided equally among n = 5 substations at the receiving 
end of the line. Let the annual effective power factor be 
cos = 0.90, the equivalent annual hours of operation be 



212 TRACTION AND TRANSMISSION. 

^ = 3500, and let the constants have the following values 
— the bracketed values being suggestive of the proper order 
of magnitude : 

P = 10. Cj, = [80]. 

w = 0.00000303. X^ = [600]. 

p2 = [0.06]. d = [0.20]. 

C2 = [0.18]. K = [0.50]. 

cs = [o.oi]. K' = [65]. 

Pi = Pp= Pt= Pa = [0.12]. Ca = [50]. 

Substituting these values, 

Kc = (0.1096 X 3000 X 350,000/0.9), 

V.oi X 10 X 3500 X .06 X .18 X 0.00000303 = 432,000, 
Kp= 3X0.12X0.2X 450,000/600 = 54, 
K/ = 3 X 0.12 X0.5 Viooo (i + Vio) = 23.75, 
Ka = 0.12 X 50 X 6 = 36. 

Substituting these values in equation (14), the economic 
voltage is 



v/' 



4^2,000 ^ , ., ,, 

^^^^^-^ - 61.7 kilovolts. 

54 + 23.75 + 36 

The American Institute of Electrical Engineers recom- 
mends as standard voltages for transmission circuits 6.6, 
II, 22, 33, 44, 66, or 88 kilovolts. Furthermore, 55-kilo- 
volt apparatus is listed by manufacturers. The problem 
in hand requires for greatest economy 61.7 kilovolts, a 
value which falls between two of those recommended. 
It is instructive to find what additional annual expense 
would be entailed in following the recommendations. The 
annual expense items for different voltages are therefore 
given in the following table. 



TRANSMISSION LINES. 

ANNUAL EXPENSES AT DIFFERENT VOLTAGES. 



213 







Kilovolts between Wires. 




Items of Annual Expense. 




















44 


55 


61.7 


66 


Conductors: 










Kc!E 


9,810 


7,860 


7,010 


6,550 


Poles and insulators: 


Kr> 


7,200 


7,200 


7,200 


7,200 


Kr>'E 

Transformers: 


2,378 


2,970 


3,330 


3,560 


Kt...^ 

Kt'E 

Auxiliaries: 


4,050 
1,044 


4,050 
1,314 


4,050 
1,467 


4,050 
1,568 


KaE 


1,582 


1,980 


2,220 


2,376 




Total annual expense 


$25,094 


$24,404 


$24,307 


$24,334 



These results show that the additional annual expense 
would be but $97 at 55 kilovolts or $27 at 66 kilovolts, and 
therefore either standard voltage should be adopted. The 
use of the higher voltage requires a somewhat smaller 
initial investment. It may be desirable in some cases 
materially to increase the operating voltage above that 
determined in this manner, in order to limit the first cost. 

72. Separation of Conductors. — The proper separation 
of the conductors of a transmission line depends largely 
upon the insulating properties of the atmosphere. If the 
voltage between two aerial conductors be gradually in- 
creased a critical voltage is reached at which a discharge of 
electricity from the conductors into the air is initiated. 
This critical voltage depends upon the sizes of the con- 
ductors and the distance between them, and upon the 
temperature and pressure of the air. When this voltage 
Is exceeded, the conductors when seen at night are sur- 
rounded by a luminous envelope of red-violet color. The 
phenomenon is termed corona, A considerable loss of 
15 



214 



TRACTION AND TRANSMISSION. 



energy results from the employment of transmission voltages 
above the critical value, this loss being termed corona loss. 
At normal pressure and temperature, the air breaks down 
and becomes convectively conductive when subjected to a 
uniform electric field strength of 78 kilovolts per inch or 
31 kilovolts per centimeter. For alternating currents these 
values apply to the maximum value of the voltage wave, 
therefore, if a sine wave is assumed the strength of air is 
31 -^ '\2 or 21.9 effective kilovolts per centimeter. The 
electric fields in the vicinity of the conductors of an ordinary 
aerial line are not uniform, for the lines of electrostatic flux 
diverge in leaving the conductors. The amount of diverg- 
ence depends upon the sizes 
■^^ ^ of the conductors and the 

distance between them. 
The distribution of potential 
between two wires, A and B, 
is typified by Fig. 90. The 
strength of the electric field 
at any point between the 
conductors is measured by 
the slope of the curve at that 
point, and is termed the volt- 
age gradient. This gradient, expressed in volts (or kilo- 
volts) per centimeter, is seen to be greatest at the wire 
surface and least midway between the wires. In conse- 
quence, it is at the surface of the wires where the break- 
down of the air must first occur, and therefore the electrical 
conditions for the starting of corona may be determined from 
the critical field intensity at the surface of the conductor. 
The physical process underlying the initiation of corona 
is termed ionization by collision. Due primarily to the 



DISTANCE 
BETWEEN WIRES 



Fig. 90. 



TRANSMISSION LINES. 215 

presence of radioactive substances on the earth, there are 
always present in the atmosphere positive and negative 
ions, each carrying a charge of 4.6 X io~^^ abstat coulombs 
or multiples thereof. Under ordinary conditions the num- 
ber present per cubic centimeter is of the order of 1000, 
and this number is inadequate to permit of appreciable 
convective conduction by the air of the problem under 
consideration. Each of these ions however, if subjected to 
an electric field intensity of sufi&cient magnitude, will acquire 
adequate kinetic energy in traversing a free path, to ionize 
a neutral air molecule with which it collides. The energy 
required to ionize a gas particle, as determined by various 
methods, is of the order 4 X io~^^ ergs. Since the free 
path of a gas particle increases directly with decrease 
of pressure at constant temperature and with increase of 
temperature at constant pressure, the value of the critical 
voltage will accordingly decrease with like proportionality. 
Numerous experiments show that the voltage gradient 
(maximum value in the case of alternating currents) at the 
surface of the wire when corona appears on a round wire 
located in air depends upon the diameter of the wire and 
the density of the air in a manner given by Peek's equation 

g =^ Ab -\- B yj — kilovolts per cm., (i) 

where D is the diameter of the wire in cm., A and B are 
constants, and 5 is a density factor. The value of this 
density factor is unity at a barometric pressure of 76 cm. 
of Hg. and a temperature of 25° C, while for any other 
pressure p cm. and other temperature f C. its value is 

76 * 273 + ^ 273 + r ^ ^ 



2l6 TRACTION AND TRANSMISSION. 

The values of the constants A and B in equation (i) 
obtained by various observers are somewhat at variance 
with each other, due probably to difficulties in measuring 
the ratio of the maximum to the effective values of the 
high alternating voltages and to the dissimilar surface con- 
ditions of the wires tested. Recent tests by Prof. J. B. 
Whitehead on wires ranging from 0.074 to 0.231 cm. 
diameter yield the following values for A and B : 



Alternating current 

' wire positive 



A 


B 


337 


12.6 


337 


II-5 


31.02 


13-5 



Continuous current , 

wire negative 

F. W. Peek uses as his most recent coefficients corresponding 
to A and B for alternating currents the following: 
A = 31.0 and B = 13.5. 

The usual experimental method for ascertaining the 
value of the critical surface gradient g is to apply a high 
voltage between a wire and a metal tube, which tube is 
placed coaxially around the wire. If the wire has a diam- 
eter D cm. and the tube a radius of b cm., and corona 
appears when the applied voltage has reached a maximum 
value of V kilovolts, then the critical voltage gradient at the 
wire surface is given by 

2V 
g = 7 kilovolts per cm. (3) 

Prof. Bennett calls this the hypothetical gradient because it 
holds for pure dielectrics permitting of elastic displace- 
ment only, but does not necessarily hold for dielectrics at 
potentials causing corona around the wire. Corona pro- 
duced by alternating currents may be detected by visibility 



TRANSMISSION LINES. 



217 



or by an electroscope, while for direct-current corona a 
galvanometer may be used as an additional method. In a 
particular experiment by Whitehead, a wire of 0.231 cm. 
diameter within a tube 6.109 cm. in diameter required 
22,500 volts (maximum value of alternating voltage) for 



o 



> 

1 60 

X 

< 



a 
< 











































\ 




















\ 


\ 




















K 


^^ 




















^^ 








1EAD_ 





























































































0.8 1.2 

WIRE DIAMETER IN CM. 

Fig. 91. 



1.6 



2.0 



the appearance of corona; the critical surface intensity from 
equation (3) is therefore 



g = 



2 X 22.5 



0.231 X 2.3026 logio 



6.109 
0.231 



= 59.6 kilovolts per cm. 



Having experimentally determined the values of g in this 
way for various values of wire diameter D and of density 
factor 5, the numerical magnitudes of the constants A 
and B of equation (i) are then ascertained. 



2l8 TRACTION AND TRANSMISSION. 

The critical surface gradients for various sized conductors 
are given in Fig. 91 as calculated from equation (i), using 
the alternating-current constants A and B as given by Peek 
and Whitehead and taking 5 = i. The curves are prac- 
tically coincident over the range of Whitehead's experi- 
ments, but separate beyond a conductor diameter of 0.2 cm. 

The gradient at the surface of a wire of diameter D cm. 
within a conducting cylinder of radius b cm. with a voltage 

V across the wire and cylinder is the same as the surface 
gradient of the same wire when used as one of parallel wires 
if the voltage from wires to neutral is equal to V and the 
distance d from center to center of the wires is equal to b. 
Then, from equation (3), the gradient for the appearance 
of corona on parallel wires is 

2V 
g = -,kilovolts per cm., (4) 

D log. ^ 

where V is the voltage from a wire to neutral in kilovolts 
(maximum value) and d is the conductor separation in cm. 
Solving this expression for V and using equation (i) there 
results as the visual critical voltage to neutral: 

gD 2d D f IY\ 2d 

V = ^log. - = -\^A8 + B >J- j log. -kilovolts. (5) 

To take care of the condition of the conductor surface, an 
irregularity factor m should be inserted in the foregoing 
equation. Its value is given by Peek as follows: 

Polished solid conductors 1. 00 

Roughened or weathered wires 0.98 to 0.93 
Cables (seven strand) 0.82 to 0.72 

Converting equation (5) to effective voltage to neutral by 
dividing by V2, reducing the logarithm to base 10, and 



TRANSMISSION LINES. 



219 



using A =31.0 and B = 13.5, there results as the effective 
voltage at which corona becomes visible: 



V = mD 



(25.25+ ii.o^-^j 



2d 
logio -j^ kilovolts. 



CORONA LIMITING VOLTAGES 

ON THREE-PHASE 

TRANSMISSION LINES 

Effective Kilovolts Between Wires 

for Various Spacings in Feet 

(Conductors Equidistant) 

WIRES 



CABLES 



120 



40 









/ 






4 


m 






-A 


V 




/// 


^ 




^ 


w 






^ 


/ 






f 




^0=0.95 
(5 = 1.0 











0.1 0.3 0.5 

WIRE DIAMETER 

(Inches) 

Fig. 92. 

















/. 














4' 


%. 












/ 




y/ 










/ 


V/ 


'h 














^ 


/ 










:^ 


^ 


/ 










^ 


^ 


/ 










A 


r 


/ 










A 




/ 












#^ 










mo = 0.85 
(5=1.0 




/ 

















(6) 

KV. 
260 

220 

180 

140 

100 



0.3 0.5 0.7 0.9 

CONDUCTOR DIAMETER (Inches) 

Fig. 93. 



1.1 



60 



A loss of power from transmission lines occurs at a voltage 
lower than that at which corona appears, the difference 
being considerable with small wires. This voltage, termed 
the disruptive critical voltage, is expressed by the following 
equation given by Peek and similar in form to equation (5) : 



£cr = gonio— 8 log, 



2d^ 



kilovolts to neutral, 



(7) 



where £0 is the disruptive voltage gradient at the conductor 
surface and may be taken as 21.9 kilovolts (effective value) 



220 TRACTION AND TRANSMISSION. 

per centimeter, and mo is the irregularity factor which is the 
same as m for wires but has values between 0.87 and 0.83 
for cables. The critical voltage between conductors is V3 
times the value of Ecr for three-phase lines and twice Ecr 
for single-phase lines. The critical voltage is lowered by 
smoke, fog, sleet, rain and snow, but is not appreciably 
affected by humidity, air velocity or wire material. 

The disruptive critical voltages between conductors 
(triangularly spaced) on three-phase lines are shown by 
the curves in Fig. 92 for solid wires and in Fig. 93 for 
cabled conductors, plotted from equation (7). When the 
three wires lie symmetrically in one plane the critical 
voltage for the center wire will be about 4 per cent lower, 
and for the outer wires 6 per cent higher than shown. 

73. Resistance of Conductors. — The resistance per mile 
of length of a conductor in which the current density is 
uniform throughout the cross section, A circular mils, at 
any temperature t degrees centigrade is 

7?, =5280^(1 + a/), 

where p is the resistivity in ohms per circular mil-foot at 
o degrees centigrade, and a is the mean temperature 
coefficient of electrical resistance; accepted values of which 
for the usual line materials being 

pa w 

Copper (hard drawn) 9.54 0.00415 0.00000303 

Aluminum (hard drawn) 15.8 0.0039 0.00000091 

The weights per circular mil-foot in pounds of copper and 
aluminum are given in the last column. 

Uniform distribution of current in conductors is realized 
in the transmission of continuous currents. In conductors 



TRANSMISSION LINES. 



221 



carrying alternating currents, the current density at the 
surface is greater than at the axis of the conductors; this 
unequal distribution of current increases with the fre- 
quency of the impressed electromotive force and manifests 
itself as an increase in resistance by rendering part of the 
conductor cross section ineffective. Fig. 94 shows the per- 



u 

CO 

-z. 


























/ 
























/ 


/ 



5 20 






















/ 


/ 






















y 


/ 






u 

O 10 

< 


















y 


/ 




















^ 


y 












z 






u 


uj 










^ 



















1.5 



2.0 



2.5 

Z 

Fig. 94. 



3.0 



3.5 



4.0 



centage increase of resistance of conductors when traversed 
by alternating currents over that when traversed by contin- 
uous currents in terms of a function 2, which is defined as 



= Vf 



where r is the radius of the wire in inches and / is the f re~ 
quency in cycles per second. 

The resistances per unit length of cables are somewhat 
greater than those of solid conductors of like cross-sectional 
areas. If there be A^ strands in a cable having a lay of i 
in n (i.e., the pitch of the strand hehces is n times their 
diameter measured along the central wire), the total re- 



222 TRACTION AND TRANSMISSION. 

sistance of the cable, assuming no current flow between 
strands, will be 

N 



R = Rr 



, n{N -i) 

I + - 



Thus, the resistance of a 19-strand cable having a lay of i in 
15 is 2.05 per cent greater than the resistance of a solid 
conductor of equal cross-sectional area. 

74. Line Inductance. — Conductors carrying a varying 
current are surrounded by a magnetic field of varying in- 
tensity. A change in the magnetic flux which encircles a 
conductor develops in it an electromotive force of self- 
induction. If the conductors carry an alternating current 
an alternating electromotive force will be induced in them, 
the magnitude of which depends upon the time rate of 

change of current, that is, its value at the instant Hs L -t- , 

at 

where L represents the inductance of the circuit and V is 

the instantaneous current value. 




Fig. 95. 



To determine the inductance L per unit length of single 
wire, consider a two-wire line carrying an alternating cur- 
rent, the conductors being of radius r and separated be- 
tween centers by a distance d, as shown in Fig. 95. The 



TRANSMISSION LINES. 223 

magnetic flux which passes through an element outside of 
the conductor of width dx and of unit axial length is equal 
to the magnetomotive force divided by the reluctance, or 

j^ 4 iri . dx 

d^i = -^ — = 2t — J 

2 ttX X 

dx 
where i is the instantaneous value of the current flowing in 
the conductors. The total magnetic flux which passes be- 
tween the wires due to the current in one of them is obtained 
by integration for values of x between d — r and r, as 

d — r 



^l = 2t J — = 2t loge - 

^ r X 



r 
d and r being both expressed in terms of the same unit. 

The magnetic flux which passes through the conductor 
material is of appreciable magnitude owing to the greater 
flux density near the wires. Assuming for simplicity that 
the current is uniformly distributed over the cross sections 
of the cylindrical conductors, then the current inside the 



x^ 



circle of radius :^ is — i, and the magnetomotive force which 



x^ 



it produces is 4 7r— i. The magnetic flux per unit length 
of the element dx is 2 ifx — - ? and since this flux is associ- 

ated with but — ths of the wire, the equivalent elementary 

magnetic flux which may be considered as linking the entire 
conductor is 

d^2 = 2Z/X— — • 

r 
Integrating for values of x between o and r, there results 

^2 = h i^- 



224 TRACTION AND TRANSMISSION. 

Hence the total magnetic flux linked with each conductor 
of the two-wire line is 

[d — r jLt"| 
2 loge — — + 2 h 

and therefore the inductance per centimeter length of the 
straight conductors, being the flux per unit current, in abso- 
lute units is 

d — r ^ jjL 

1 = 2 iOQ-^ h - centimeters. 

r 2 

By reduction, the inductance per mile for a single copper or 
aluminum wire becomes 



-[ 



d — r "I 

741 logio h 80.5 10-^ henries. 



For 19-strand and 7-strand cables the constant 80.5 should 
be replaced by 89 and 102 respectively. 

75. Hyperbolic Functions. — Many numerical calcula- 
tions in Electrical Engineering are greatly facilitated by 
the use of hyperbolic functions, just as are calculations in 
mechanics by the use of circular functions. The use of 
the former is as simple as that of the latter and the rela- 
tions which exist between the functions of each type are 
almost identical, the transformation formulae seldom differ- 
ing from each other in more than sign. Hyperbolic func- 
tions are especially useful in treating the problems arising 
in connection with transmission lines. 

In Fig. 96 consider the rectangular hyperbola HH and 
the circle CC concentric with O as a center. Since OA 
equals the radius, r, of the circle, yJOA is the circular sine 
of the angle 6 by conventional definition. Similarly yn/OA 
is, by definition, the hyperbolic sine, or, as it is commonly 
expressed, the sink of the corresponding magnitude. Al- 
though the circular functions are usually specified in terms 



TRANSMISSION LINES. 



225 



of the angle, d, included between the axis of abscissae and 
the radius vector through any point, Pcj of the circle, they 
might equally as well be specified by twice the area AOPc of 
the circular sector which corresponds with this angle, if the 




Fig. 96. 

radius were unity. This will become evident if it be con- 
sidered that the circular sectorial area Uc = — 7rr^, whence 

2 TT 

2 
6 = —Uc] that is, 6 varies directly with u^. The hyperbolic 

functions are not specified by the angle d but by twice the 
hyperbolic sectorial area AOPh = Uh. Referring to a circle 
of unit radius, by definition Xc/OA = cos d = cos 2 u^ and 
yjocc = tan 2 w^; similarly Xh/OA = cosh 2 Uh and yh/x^ 
= tanh 2 Uh, the final h signifying hyperboHc functions. 
The relations which exist between the coQrdinates of any 



226 TRACTION AND TRANSMISSION. 

point, Ph, on the hyperbola and the corresponding sectorial 
area Uh may be derived from the equation of the equilateral 
hyperbola, x^ — y'^ = r^. The area of the sector OAPn is 
Uh = area of triangle OQPh — area of segment AQPh 

or un = ^^^- r^ydx 

2 ^ r. 



2 
Xhjh 



-J^'Vx' 



r^ dx 

2 



= - log. 



Xh + yh 



2 r 

Therefore x^±% ^ ^^ ^^^ 

r 

Since the equation of the hyperbola may be written as 

r'^ = {x-\- y) {x - y), 

, X -\- y r 

whence = ? 

r X — y 

^^-y^ = e '' (2) 

By adding (i) and (2) 

^ = lLr^ +,- r^]. (3) 

r 2 ^ / 

In general, dropping the subscript of x, making the radius 

r = I, and letting u = -^j 

a; = i(6" + 6-^) = COShz^. . (4) 

By subtracting (2) from (i) and expressing in general form 

3^ = !(€"- 6-^) =sinhzi, (5) 

and dividing (5) by (4), 

y sinhw ^ , /^n 

^ = — \ — = tanh u. (0; 

X cosh li 



TRANSMISSION LINES. 227 

The ratio of the areas, u = 2 Uh/r^^ is termed the argument, 
which specifies the functions. 

For large values of tc the second exponential terms in 
equations (4) and (5) vanish and sinh u = cosh u while 
tanh z/ = I . 

Relations between the Functions. — The following useful 
formulae, showing some of the relations existing between 
the hyperbolic functions, may be derived readily from the 
properties of the hyperbola or by substitution or transfor- 
mation. 

cosh^u — sinh^z^ = 1. (7) 

sinh {u zb v) = sinh u cosh v ± cosh u sinh v, (8) 

cosh (u iLv) = cosh u cosh v zb sinh u sinh v. (9) 

sinh 2u = 2 sinh u cosh u. (10) 

cosh 2U = cosh^z/ + sinh^^. (11) 

cosh z^ it sinh ^ = e"^"". (12) 

Differential Coefficients, — By successively differentiating 
equations (4) and (5) there results 

J sinh z^ e"" + €"■'' 1 d^sinhu . , . . 

; = = cosh u: -— — = smh u, (ly 

du 2 ' du^ ^ V 0/ 

d cosh u e^ — e'"^ . , d" cosh u , . . 

= — = smh u\ — I — = cosh u. (14) 

du 2 dtr 

This repetition of the functions, after two successive 
differentiations, is the basis of their utihty in problems of 
decay or attenuation. 

Tables. — An excellent set of tables and formulas relating 
to this subject is published by the Smithsonian Institution 
of Washington in Publication No. 187 1 bearing the title 
^^ Hyperbolic Functions." The numerical values of cosh 
and sinh for arguments from 0.00 to 8.45 are given in the 
following table. 



228 



TRACTION AND TRANSMISSION. 









HYPERBOLIC 


FUNCTIONS. 






u. 


sinh M. 


cosh u. 


u. 


sinh u. 


cosh u. 


u. 


sinh u. 


cosh u. 


o.oo 


. 0000 


I . 0000 


0.50 


O.5211 


1. 1276 


1 .00 


1-1752 


I -5431 


OI 


0100 


0001 


51 


5324 


1329 


05 


2539 


6038 


02 


0200 


0002 


52 


5438 


1383 


10 


3356 


6685 


03 


0300 


0005 


Sd> 


5552 


1438 


15 


4208 


7374 


04 


0400 


0008 


54 


5666 


1494 


20 


5095 


8107 


05 


0500 


0013 


55 


5782 


1551 


25 


6019 


8884 


06 


0600 


0018 


56 


5897 


1609 


30 


6984 


1.9709 


07 


0701 


0025 


57 


6014 


.1669 


35 


7991 


2-0583 


08 


0801 


0032 


58 


6131 


1730 


40 


I 9043 


1509 


09 


0901 


0041 


59 


6248 


1792 


45 


2.0143 


2488 


10 


1002 


0050 


60 


6367 


1855 


50 


1293 


3524 


II 


II02 


0061 


61 


6485 


1919 


55 


2496 


4619 


12 


1203 


0072 


62 


6605 


1984 


60 


3756 


5775 


13 


1304 


0085 


63 


6725 


2051 


65 


5075 


6995 


14 


1405 


0098 


64 


6846 


2119 


70 


6456 


8283 


15 


1506 


0113 


65 


6967 


2188 


75 


7904 


2.9642 


16 


1607 


0128 


66 


7090 


2258 


80 


2.9422 


3-1075 


17 


1708 


0145 


67 


7213 


2330 


85 


3.IOI3 


2585 


18 


181O 


0162 


68 


7336 


2402 


90 


2682 


4177 


19 


I9II 


0181 


69 


7461 


2476 


95 


4432 


5855 


20 


2013 


0201 


70 


7586 


2552 


2.00 


6269 


7622 


21 


2II5 


0221 


71 


7712 


2628 


05 


3.8196 


3-9483 


22 


2218 


0243 


72 


7838 


2706 


10 


4.0219 


4.1443 


23 


2320 


0266 


73 


7966 


2785 


15 


2342 


3507 


24 


2423 


0289 


74 


8094 


2865 


20 


4571 


5679 


25 


2526 


0314 


75 


8223 


2947 


25 


6912 


4-7966 


26 


2629 


0340 


76 


8353 


3030 


30 


4-9370 


5-0372 


27 


2733 


0367 


77 


8484 


3114 


35 


5-1951 


2905 


28 


2837 


0395 


78 


8615 


3199 


40 


4662 


5569 


29 


2941 


0423 


79 


8748 


3286 


45 


5-75IO 


5-8373 


30 


3045 


0453 


80 


8881 


3374 


50 


6.0502 


6.1323 


31 


3150 


0484 


81 


9015 


3464 


55 


3645 


4426 


32 


3255 


0516 


82 


9150 


3555 


60 


6.6947 


6.7690 


2>2> 


3360 


0549 


^3 


9286 


3647 


65 


7.0417 


7-1123 


34 


3466 


0584 


84 


9423 


3740 


70 


4063 


4735 


35 


3572 


0619 


85 


9561 


3835 


75 


7-7894 


7.8533 


36 


3678 


0655 


86 


9700 


3932 


80 


8.1919 


8.2527 


37 


3785 


0692 


87 


9840 


4029 


85 


8.6150 


8.6728 


38 


3892 


0731 


88 


0.9981 


4128 


90 


9.0596 


9. I 146 


39 


4000 


0770 


89 


I .0122 


4229 


95 


9.5268 


9-5791 


40 


4108 


0811 


90 


0265 


4331 


3.00 


10.0179 


10.0677 


41 


4216 


0852 


91 


0409 


4434 


05 


10.5340 


10.5814 


42 


4325 


0895 


92 


0554 


4539 


10 


11.0765 


11.1215 


43 


4434 


0939 


93 


0700 


4645 


15 


II .6466 


11.6895 


44 


4543 


0984 


94 


0847 


4753 


20 


12.2459 


12.2866 


45 


4653 


1030 


95 


0995 


4862 


25 


12.8758 


12.9146 


46 


4764 


1077 


96 


I144 


4973 


30 


13-5379 


13-5748 


47 


4875 


1125 


97 


1294 


5085 


35 


14.2338 


14.2689 


48 


4986 


1174 


98 


1446 


5199 


40 


14.9654 


14.9987 


49 


5098 


1225 


99 


1598 


5314 


45 


15 -7343 


15.7661 



TRANSMISSION LINES. 



229 



HYPERBOLIC FUNCTIONS. 



u. 


sinh u. 


\ cosh u. 


M. 


sinh u. 


cosh M. 


3-So 


16.5426 


16.5728 


6.00 


201.7132 


201.7156 


55 


17-3923 


17.4210 


05 


212.0553 


212.0577 


60 


18.2855 


18.3128 


10 


222.9278 


222.9300 


65 


19.2243 


19.2503 


15 


234.3576 


234-3598 


70 


20.2113 


20.2360 


20 


246.3735 


246.3755 


75 


21.2488 


21.2723 


25 


259.0054 


259-0074 


80 


22.3394 


22.3618 


30 


272.2850 


272.2869 


S5 


23 4859 


23-5072 


35 


286.2455 


286.2472 


90 


24.6911 


24.7113 


40 


300.9217 


300.9233 


3 95 


25-9581 


259773 


45 


316.3504 


316.3520 


4.00 


27.2899 


27.3082 


50 


332.5700 


332.5716 


05 


28.6900 


28.7074 


55 


349.6213 


349-6228 


10 


30.1619 


30.1784 


60 


367.5469 


367.5483 


15 


31.7091 


31.7249 


65 


386.3915 


386.3928 


20 


33-3357 


33-3507 


70 


406.2023 


406.2035 


25 


35-0456 


35-0598 


75 


427.0287 


427.0300 


30 


36.8431 


36.8567 


80 


448.9231 


448.9242 


35 


38.7328 


38.7457 


85 


471.9399 


471.9410 


40 


40.7193 


40.7316 


90 


496.1369 


496.1379 


45 


42.8076 


42.8193 


6.95 


521.5744 


521.5754 


50 


45.0030 


45.0141 


7.00 


548.3161 


548.3170 


55 


47-3109 


47-3215 


05 


576.4289 


576.4298 


60 


49-7371 


49.7472 


10 


605.9831 


605.9839 


65 


52.2877 


52.2973 


15 


637.0526 


637.0534 


70 


54-9690 


54-9781 


20 


669.7150 


669.7157 


75 


57.7878 


57-7965 


25 


704.0521 


704.0528 


80 


60.7511 


60.7593 


30 


740.1497 


740.1504 


85 


63.8663 


63.8741 


35 


778.0980 


778.0986 


90 


67.1412 


67.1486 


40 


817.9919 


817.9925 


4-95 


70.5839 


70.5910 


45 


859.9313 


859.9318 


5.00 


74-2032 


74.2099 


50 


904.0210 


904.0215 


05 


78.0080 


78.0144 


55 


950.3711 


950.3716 


10 


82.0079 


82.0140 


60 


999-0976 


999.0981 


15 


86.2128 


86.2186 


65 


1050 


.323 


20 


90.6334 


90.6389 


70 


1 104 


• 174 


25 


95 ■ 2805 


95.2858 


75 


1160 


.780 


30 


100.1659 


100.1709 


80 


1220 


.301 


35 


105.3018 


105.3065 


85 


1282 


.867 


40 


110.7009 


110.7055 


90 


1348 


.641 


[ ^^ 


116.3769 


116. 3812 


7-95 


1417 


.787 


50 


122.3439 


122.3480 


8.00 


1490 


■479 


55 


128.6168 


128.6207 


05 


1566 


.698 


60 


135-2114 


135-2150 


.10 


1647 


.234 


65 


142.1440 


142.1475 


15 


1731 


.690 


70 


149.4320 


149-4354 


20 


1820 


•475 


75 


157-0938 


157.0969 


25 


1913 


-813 


80 


165.1482 


165.1513 


30 


201 1 


-936 


ss 


173-6158 


173.6186 


35 


2115 


.090 


90 


182.5173 


182.5201 


40 


2223 


■533 


95 


191.8754 


191.8780 


45 


2337 


-537 



16 



230 TRACTION AND TRx\NSMISSION. 

76. Line Capacity. — To determine the capacity of a 

transmission line, consider two wires of indefinitely small 

diameters placed d' centimeters apart and having respec- 

P tively charges of + ? and — q 

y^'^^ units per centimeter length of 

y^ ^\,^^ conductor. The intensity of 

— ¥^ -1 "^"f the electric field at a point P, 

^^ _^j ^, pjg ^y^ distant ri cm. from one 

^^s. 97. wire and r^ cm. from the other, 

that is, the electrostatic flux per unit area of equipo- 

tential surface or force exerted upon a unit positive charge 

at this point due to the charge on wire A alone, is 

2 irri Ti 

and that due to the charge on wire B alone is 
^ - 4 7rq q 

2 7rr2 ^2 

Representing the potential at the point P due to the 
charge on Ahy Va^ and that due to the charge on B by F^, 
it follows from the definition of potential that 

dVA _ 2q 

dri kri 

, dVB 2_q 

and — - = - -^ 

dr2 kr2 

where k is the permittivity or specific inductive capacity 
of the dielectric. If the potentials at the point midway 
between the two very small wires due to their charges be 
respectively Va and Vb\ then the potential difference 
between P and is the sum of 



TRANSMISSION LINES. 23 1 

and Vb- Vb' = f - ^ dr2 = - ^log.— • 

Or, since for the point 0, Fa^ + Fb' = o, the potential at 
P due to the charges on both wires is 

Va+Vb =V=^log/-^' (i) 

For any point to be on the equipotential surface which 
passes through the point P, the ratio of its distances from 
B and A respectively must be constant. The locus of a 

point P which moves so that — is constant is a circle, and 

ri 

if C be its center and r its radius, then ] 

CAXCB= r\ (2) 




Fig. 98. 



From Fig. 98, which is drawn in accordance with equa- 
tion (2), it appears that triangles ACP and BCP with the 

CA r 

common angle at C are similar, since from (2) — = — — • 

r CB 



Therefore 












AP BP 
CA CP 


or 


CA 


r 


and 


CA 
r 









232 



TRACTION AND TRANSMISSION. 



which shows that — is constant whatever the position of P 

on the circle. Consequently the equipotential surfaces re- 
sulting from the charges on the two wires A and B are 
cylindrical in shape and are not coaxial with those wires; 
furthermore, the axes of such cylinders of different radii are 
not coincident. The radius of the zero potential surface 
which passes through the mid-point must be infinitely 

large (for ~ = ij, and therefore this surface is a neutral 

plane which bisects the line AB Sit right angles. All the 
equipotential surfaces to the left of this plane surround A 
and those to the right surround B. 

Consider two equipotential surfaces surrounding the 
wires A and B to be replaced by solid cylindrical conductors 




of radii ai and a2 respectively, Fig. 99, and carrying charges 
respectively oi + q and — q units per centimeter length of 
conductor. Such substitution does not alter the potential 
or electric flux distribution beyond these surfaces. The 
potentials of the wires are respectively 



Fi = 



2q , BM 

T'^^^am 



TRANSMISSION LINES 233 

J Tr 2g, AN 

and F2= -yloge^. 

Therefore their potential difference is 

r. ^r rr 2 0, BM ' AN 



1 


._., ..,_ ^-^.^^.^^ 


But from the 


figure 




5M 51f' 5Ci 




AM AM' ax 


and 


AN AN' AC2 
BN BN' 02 ' 


also 


AC2 X BC2 = 02^ 


consequently 




Furthermore, 






BCi (BCi - d') = ai2 


and 


5C2 (5C2 + d') = 02^; 


whence 


2 V 4 



and £C2 = - - + y/a2' + 



d'^ 



(3) 



Therefore the capacity per centimeter length of Hne having 
two cylinders of radii ai and ^2 as conductors is, from (3), 

k 



If both wires have the same diameter, r = ai = ag, and 
the capacity is 



234 



TRACTION AND TRANSMISSION. 
k 



C = 



2 log. 



v-(f:) 



2 r 



_V \2r/ 2Y 



(4) 



Representing the distance between conductor axes by J, it 
is seen that 



whence d! = Vt/^ — 4 /'^^ 

Therefore the capacity of a transmission line having con- 
ductors of r centimeters radius is 



C 



2 log. 






or, letting — = ;w, this becomes 
2 Z' 



C = 



+ Vm^ ^ 1 1 4 log€ (w + '>'^^ "" i) 



r m + yw? ^ I "I 

loge / , 

Lw — Afm^ — I J 



This may also be expressed as 
k 



C = 



electrostatic units. 



4 cosh ^ w 

Reducing to microfarads per mile, the capacity of either 
wire with respect to the neutral plane is 

0.0388 k 



C = 



'^^42^+^'^^""'] 



(5) 



TRANSMISSION LINES. 235 

0.0895 k 
or C = v . (6) 

cosh~^ — 
2r 

By neglecting the unity term under the radical of equation 
(5), it assumes the approximate and widely known form: 



0.0388 k 
logio- 



C=^^. (7) 



In applying these equations and that for inductance 
(§ 74) to 3-phase circuits where the conductors are not 
equidistant, the effective spacing d is to be taken as the 
cube root of the product of the three conductor spacings. 

77. Equations of Wave Propagation along Wires. — Any 
polyphase transmission line can be resolved into separate 
single-phase single-wire circuits with imaginary perfectly 
conducting ground return paths. Thus, the voltage on a 

representative single-wire circuit of a three-phase trans- 

77 
mission line with E volts between wires is — -^ , which is the 

V3 
voltage from one conductor to neutral. Such a line trans- 
mits one-third of the total power. It is therefore only 
necessary to consider the current and voltage distribution 
on a single-wire circuit. 

Consider the element ds of a uniform line with a per- 
fectly conducting ground return circuit, at a distance ^ from 
the end upon which an alternating electromotive force is 
impressed, as shown in Fig. 100. A current will flow 
through the conductor, which at a given instant / at the 
element ds may be represented by r, and that in the ad- 
jacent elements by r + dT and r — dl\ the latter refer- 
ring to the next adjacent element more remote from the 
generator. Let E^ be the potential at this instant of the 



236 TRACTION AND TRANSMISSION. 

line with respect to the earth at the element ds, and let 
the potentials of the adjoining elements be £' + dE^ and 
E^ — dE^ respectively. Let R, L, and C in homologous 

; r^ \\'-d\' 
H 




Ih 



Fig. 100. 

units represent respectively the uniformly distributed re- 
sistance, inductance, and capacity per unit length of the 
line. 

The difference of potential between the two ends of the 
element ds is dE^, and this must be equal to the sum of the 
resistance and inductance reactions of the elementary line 
section occasioned by the current /'; consequently for this 
element 

Lds^^ + RdsV = -dE 
dt 

T dl . „ w dE f V 

Since the line has capacity with respect to the earth, it 
takes a charging current; and in addition a slight leakage 
current may flow. Therefore the current which does not 
continue beyond the element ds, but which flows from the 
line to ground under the voltage £', is 

-dr = ^(E'Cds) + E'gds, 
at 



TRANSMISSION LINES. 237 

where g is the leakance or the reciprocal of the insulation 
resistance per unit length of line. Then 

Diflferentiating (i) with respect to time and (2) with 
respect to distance, there result respectively 

df dt ~ dAds J ds\di ) 

dH' ^ d /dE'\ , dE' 

Substitution of the former in the latter equation gives 
dH' ^^dW ^ j.^dl' dE' 

and replacing the last term by its equivalent from (i) 
there is obtained the differential equation of current propa- 
gation along a line as 

CL% + (RC + iL)'-L = '^-R,I'. (3) 

Similarly, by differentiating (i) with respect to distance 
and (2) with respect to time, and combining the resulting 
expressions, there results the differential equation of volt- 
age propagation as 

Cif + (i?C + ,L)f = f'-*,E'. (4) 

Equations (3) and (4) are identical as to f and E^, and 
their solution indicates the current and voltage values at 
the point distant s from the generator at the time t in 
terms of the line constants. This general equation refers 
to any circuit with distributed capacity and inductance, 



238 TRACTION AND TRANSMISSION. 

and its solution is of importance in telephonic and power 
transmission problems. 

78. Attenuation and Wave-Length Coefficients. — The 

solution of the equation of wave propagation may readily 
be effected by not considering the short unsteady period 
immediately following the application of voltage to the line, 
for then the solution may bie simplified by the introduction 
of the complex quantity which results in the elimination of 
the time variable. The resulting expressions are complex 
quantities and their interpretation must be made accord- 
ingly. 

Introducing the quadrantal operator, j =V— 1, and 
counting the distance s positive from the receiving end of 
the line, equations (i) and (2) of § 77 for the steady state 
may be written* 

^={R+j<.L)I^ (i) 

and ^=(g+icoC)£., (2) 

where E^ and 7^ represent the maximum (or effective) 
values of electromotive force and current at any point on 
the circuit, {R + jcoL) is the conductor impedance, and 
is + J^C) is the dielectric admittance. Differentiating 
either of these expressions and substituting the other in the 
result yields respectively 

^ = (i? +j<.L) (g +jcoC) £„ =y'E„, (3) 

and ^ = (^ +>^) (S +i'^C) /„ = y/„, (4) 

* See p. 74, Alternating Current Machines (1908) by Sheldon, Mason, 
and Hausmann. 



TRANSMISSION LINES. 239 

where y^ ={R + jo)L) {g + jcoC) for convenience. Equa- 
tions (3) and (4) are identical equations as to £^ and /^ 
and differ only in the terminal conditions, consequently the 
solution of one will suffice. 

Considering equation (4) and multiplying through by 

2 -7^ J there results 
as 

ds ds^ "^ ds 

which when integrated becomes 

Replacing the constant of integration Ci by y^c<^y where C2 
is also a constant, and separating the variables, there results 

= y as. 



Integration yields 

loge [C3 dm + V77T^)]= 7^, 

where c^ is another constant of integration. Writing in 
exponential form, this equation becomes 



= (/^ + VlJ + C2') 



Cz. 



2ys ys 

t , X o X c 



Squaring, I J + ^2- = -y + / J - 2 I^ — y 

Cz Cz 

^2ys ^ys 

or —J - C2^ = 2/^—; 

Cz Cz 

whence 



-ys f. 2r ^—ys 

Ir.=—- ^^^^ =^6- -Be-y% (5) 

2^3 2 

where the two constants are A = and B = -^— ^ • 

26:3 2 

Since the exponential coefficient 7 is the square root of 



240 TRACTION AND TRANSMISSION. 

the product of two complex numbers, it also is a complex 
quantity, and may be written 

7 = i8+i«, (6) 

where jS and a are its two rectangular components. Then 

or (/32 -a^) + 2 ja^ = (Rg - c^'CL) +j (gcL +coi?C). 
This equation can be true only if 

«2 _ ^2 _ ^2CL _ Rg^ 

and if 2aP = co(i?C + gL). 

These are simultaneous equations which can be solved for 
a and /3. Thus, substituting the value of a from the latter 
in the former gives the biquadratic 

P' + {c'LC - Rg) ^'-- (RC + giy = o; 

4 
whence 

lS2 = - ^^JZ^ ± I VicJ'LC-RgY + co^ {RC + gLf 

2 2 

and 

^ = \/i [VCco^C^ + g') {R' + co^L^l - co^ZC + i?^] ; (7) 
similarly 

a = \/i [V(a;^C2 + g2) (7^2 + ^2^2) ^ ^2^C - i?g]. (8) 

The constant 13 is called the attenuation coefficient^ and a is 
called the wave-length constant. These constants give the 
value of 7 in equation (5) for the current at any point of 
the line. 

79. Current and Voltage Distribution on Lines. — Ap- 
plying hyperbolic functions to equation (5) of the fore- 
going paragraph for the current on a line at a point distant 
5 from the receiving end, there results 

Im = A (cosh ys + sinh ys) — B (cosh ys — sinh 75). 
= {A - B) cosh 7^ + (^ + B) sinh 7^. (i) 



TRANSMISSION LINES. 241 

The voltage at the same point is found by differentiating 
(i) with respect to distance and substituting—^ in equa- 
tion (2) of § 78. Since 

-7- cosh ys = y sinh 7^ and -7 sinh ys = y cosh 75, 

as ds 

there results 

Em = -^--4^ [(^ - ^) sinh 7^ + (^ + B) cosh 7^]. (2) 

The constants A and B of equations (i) and (2) may be 
determined from the conditions at the receiving end of the 
line. Let Er and /^ be the maximum (or effective) values 
of the voltage and current at this terminal. Then for 
5 = 0, since cosh (o) = i, and sinh (o) = o, 

Ir = A - B 

and E^ = ^-^(A+B), 

Substituting these values in (i) and (2) and remembering 
that the complex quantity 

72 = (/5 +jay = (R +jo,L){g +jo,C), 
there results: 

Im = It cosh 7^ + Er p , . ^ siuh ys (3) 

and Em = Er cosh ys + Ir — —^-7, sinh ys. (4) 

When s is reckoned from the generator toward the receiving 
end of the line, these equations become 

/3 -\' ja 
Im = Ig cosh ys - Eg ^ 7co "Z^^^^ ^^ ^^^ 



242 TRACTION AND TRANSMISSION. 

P + ja 
and Em = Eg cosh 7^ — Ig . ^ sinh ys. (6) 

The hyperboHc functions of the quantity 7 may be 
expanded by using cosh ju = cos u and sinh ju = j sin u; 
thus 

cosh 7^ = cosh {^s +jas) = cosh fis • cos as +j sinh ^s • sin as 
and 

sinh ys = sinh jS^ • cos as +j cosh jSj • sin as. 

The terminal conditions in any special problem are usu- 
ally specified, the voltage being considered the reference 
phase. In the present notation for vector rotation a cur- 
rent leading the voltage is written ii +7V2 and a lagging 
current is represented by ii —7^2. 

From equation (5) it is seen that for an infinitely long 
line, s = 00, on which the current at the inaccessible end 
is zero, Im = o, 

which, when substituted in the same equation, gives the 
current, at a point distant s from the generator end of 
such a line, as 

Im = Ig (cosh ys — sinh ys) = Ige~^\ 

Similarly Em = Ege'^' = Ege'^'e-^'^^ 

The exponential function with the imaginary exponent 
may be written in the trigonometric form by means of the 
expression ^^jas ^ ^^^ ^^ ^ j ^^^ ^^^ 

T E 

whence -y^ = -^r = €~^* (cos as — j sin as)y (7) 

wherein a is the delay in phase in radians per unit length of 
line (mile). If a length of line r miles be chosen so as to 



TRANSMISSION LINES. 243 

contain exactly n wave-lengths, then 21^71 — ar, and the 

wave length is 

. r 2 IT 

X = - = miles. 

n a 

As the frequency / of the impressed electromotive force is 

— cycles per second, the velocity of wave propagation 

2 TV 

W^ll be CO 2 TT CO 

V = f\ = — • — =— miles per sec. 
2 IT a a 

The expression for a in terms of the line constants is given 
in § 78. For a perfectly insulated resistanceless line ^ = o, 
R = Oy a = coaLC, and the velocity of wave propagation 

I 
^ — ~Jf^y is that of light, namely 3 X 10^^ centimeters 

per second, or 186,000 miles per second. 

80, Regulation. — The voltage regulation of a trans- 
mission line is the ratio of the voltage variation at the 
receiving end between no load and full non-inductive load 
to the full-load voltage at the same end of the line for 
constant impressed voltage at the other end. 

When the transmission line is open-circuited at the re- 
ceiving end, the current, Ig^, entering it at the generator, 
called the charging current, is obtained from equation (5) 
of the preceding article iox s = S = total length of the 



line, by placing 


Im = 0. 


Then 


13+ja sinh75 
'R+jo^L\oshyS 


Since 


sinh75' . 1 
' - tanh yS, 
cosh 75 


this becomes 


^-^■/^A-"^^- 



(l) 



244 TRACTION AND TRANSMISSION. 

Substituting this value for Ig in equation (6) of § 79, there 
results the voltage at any point distant s from the gen- 
erating end of the line as 

E^ = Eg (cosh 7^ — sinh ys • tanh yS), (2) 

and the voltage at the receiving end for ^ = 5 as 

Er^ = Eg (cosh yS — sinh yS • tanh 76*), 
or, since cosh^75 — sinh^75 = i, 

The regulation of the transmission line is then expressed 
as 

Regulation = ^-^^^ = ^^^ech^^-E. ^^^ 

Er Er 

81. Numerical Illustration. — Let it be required to trans- 
mit 10,000 kilowatts at 60 cycles over a three-phase aerial 
transmission line 300 miles long, employing stranded alu- 
minum conductors 0.63 inch in diameter of area 0.236 
square inch, triangularly spaced with 9 feet interaxial dis- 
tance. The voltage at the receiving end of the line is to 
be 100,000 volts between conductors, and the power factor 
of the load is 85 per cent lagging. Determine the voltage 
to be impressed on the line, the entering current, the effi- 
ciency of transmission, the voltage regulation of the line, 
and the charging current. 

The constants per mile of a representative single circuit 
with a perfectly conducting ground return path and carry- 
ing one-third of the total energy, are 
R = 0.30 ohm, 
L = 0.00196 henry, 
C = 0.0153 X 10""^ farad, 
g = practically zero. 



TRANSMISSION LINES. 245 

The current per single circuit (or per wire) at the load 

end is 

J 10,000,000 ^„ 

Ir = ■ ■ = 08.0 amperes, 

^^ 100,000 ^^ o 
3 X 7— X 0.8s 

V3 
or /, = 68.0 [0.85 -isin (cos-io.85)] = 57.8 - 35.87; 

the voltage at the receiving end, namely ~ — or 57,700 

^3 
volts per phase, being considered the datum phase. 

The attenuation and wave-length constants per mile for 

a frequency of 60 cycles (whence co = 377) are respectively 



^ = V 2,88 (V0.090 + 0.5476 — 0.74) X io~^ =0.000412 

and a = V2.88 (0.799 + 0.740) X io~^ = 0.00210. 

The hyperbolic and circular functions respectively of 13s 
and as for the total length of the transmission line are 

cosh (0.1236) = 1.00765 cos (0.630) = cos 36° 6'= 0.8080 
sinh (0.1236) = 0.1239 sin (0.630) = 0.5892. 

The current at the generator end of the line may then be 
obtained from equation (3) of § 79 as 

^^ = (57-8 - 3S-8i)(i-oo765 X 0.8080 + 0.1239 X 0.58927) 

+ 57'7 r''^^^i J (0-1239X0.8080+ 1.00765X0.58927), 

Vo.30+0.747/ 

or 

Ig = (57.8 - 35.87) (0.8142 + 0.07307) 

+ 90.5 (1.678 + 0.3257) (o.iooi + 0.59377) 

= 49.67 - 24.937 + 90.5 (- 0.0249 + 1.0277) 

= 47.42 + 68.017 amperes, 

and the current from the generator per wire is 82.9 amperes. 
17 



246 



TRACTION AND TRANSMISSION. 



Similarly the voltage at the generator end of the trans- 
mission line is 

^. = (57-8 - 35.8i)^^^|^^'(o.iooi +o.5937i) 

+ 57.700 (0.8142 + 0.0730 j) 
= (57-8-35-8i) (0.364-0.07157) (o. 1 001 +0.5937 i) 10' 

+ (46.95 + 4-2ii) 10^ 
= (12.04 + 9-23i + 46-95 + 4-21 i) 10^ 

= 58,990 + 1 3. 440 i, 

and the voltage per single circuit to be impressed on the 
line in order to have 57,700 volts per phase at the receiving 
end is 60,490 volts. 



\ 




Fig. loi. 

The vector diagram, Fig. loi, exhibits the phase rela- 
tions of the voltages and currents at the ends of the line. 
It is seen herefrom that the current at the generator end 
leads the voltage at the same place by the angle 55° 7' — 
12° 50', or 42° 17'. 

The efficiency of transmission at full load is 

57.700 X 57.8 



60,490 X 82.9 cos (42° 17') 



7. = 0.899, or 89.9 per cent. 



TRANSMISSION LINES. 247 

Since cosh yS = 0.8142 + 0.07307, the voltage at the 
receiving end on open circuit for the same impressed E,M.F. 
at the generator end is 

^ 58.990 + 134407 , 

^- = a8i42l^"^:o73^ = 73'35o + 9,9307, 

and the absolute value is 74,900 volts. Consequently the 
voltage regulation of the transmission line for 85 per cent 
power factor is 

73>900 - 57 700 ^ Q, ^Q , 

— = 0.281, or 28.1 per cent. 

57700 

The charging current per single circuit or per wire is 
obtained from equation (i) of § 80 as 

.X /0.412+2.1 7\/o.iooi+o.5937 A 

= (i38.54-3i.5j) (l. 678+0.325 J) (0.1248+0.4767) 
= - 18,8 + 118.07, 

and the absolute value is 119. 5 amperes, and leads the volt- 
age Er by 99° 3'. Therefore the charging current at the 
generating end of the line leads the voltage at the same 
place by the angle 99° 3' — 12° 50', or by 86° 13'. 

82. Corona Loss. — It is found by experiment that the 
corona loss on a transmission line is proportional to the 
square of the excess voltage over the critical disruptive 
value, and is dependent upon, the frequency, the size of the 
conductors, the distance between them, and the density of 
the air. The loss per mile in watts on a single conductor 
under fair weather conditions is given by Peek as 

f + 2s ro 

p = s^9^-^Y^ \^ (^- - ^-)^ (I) 



248 TRACTION AND TRANSMISSION. 

where Em is the voltage (effective value in kilovolts) from 
conductor to neutral, Ecr is the effective value of the critical 
disruptive voltage in kilovolts to neutral (§ 'J2)j f is the 
frequency in cycles per second, 5 is the air density factor 
(§ 72), D is the conductor diameter in inches, and d is the 
distance between centers of conductors in inches. 

This expression shows that the corona loss increases very 
rapidly as the voltage is raised beyond the critical voltage 
Ecr- It is not desirable nor economical to operate trans- 
mission lines above the disruptive critical voltage because 
of the production of noise and luminous discharge and of 
chemical action at the conductor surface. The corona loss 
is increased by smoke, fog, rain, sleet and snow; in order 
to approximate the loss during storms consider Ecr as 80 
per cent of its fair-weather value as given by equation (i). 

The method of measuring corona loss is by means of a 
wattmeter, the current coil of which is connected directly 
in the transmission line at the neutral, which is grounded, 
and the potential coil of the wattmeter is connected to the 
high-potential transformer coil. 

An important consideration arises when the distant end 
of a long transmission line is open-circuited, for the voltage 
at every point on the line increases, and the potential over 
a considerable portion of the circuit exceeds the critical 
voltage, and consequently a loss of energy ensues. This 
loss begins at that point where the voltage £0 is just equal 
to the critical value E^r, and becomes greater and greater 
as the far end is approached. The voltage at any point on 
an open-circuited line is given by equation (2) of § 80. 
By substituting various values of s therein, and plotting 
the corresponding values of Eq in terms of distance, a 
voltage-distribution curve for the particular line will result. 



TRANSMISSION LINES. 



249 



From this voltage-distance curve can be seen the distance, 
^0, from the generator end of the transmission Hne at 
which corona loss begins. Of course, this equation might 
be solved for ^o, but not knowing the phase of voltage £0 at 
the end of this part of the circuit, this plan leads to diffi- 
culty in the solution of actual problems. 

In order to determine the total corona loss on an open- 
circuited single conductor 
of length S, consider an 
element ds of the circuit, 
distant s miles from the 
point ,^0 where corona loss 
begins, for which the ex- 
cess voltage is Em ~ Ecr 
kilovolts; Fig. 102. The ^'^' ^°^- 

power loss over this elementary line section in watts is 

dP = K {Era - E,rY ds , 

where K replaces the terms of equation (i) not included in 
the parenthesis; and over the entire distance I = S — Sq 
the loss is 




= K f {Em- 



EcrY ds. 



Applying equation (2) of § 80, 

Em = Ecr (cosh 7^ — siuh 7^ tanh 7/) ; 



therefore 
P = KEJ 



I (cosh ys — 



sinh ys tanh yl — lY ds, 



= KEcr^ I cosh^ ys ds — 2 tanh yl I sinh 7^ cosh ys ds 



•^0 



I sir 

Jo 



cosh ys ds + tanh^ yl I sinh^ 7^ ds 



I sir 
Jo 



250 TRACTION AND TRANSMISSION. 

+ 2 tanh 7/ I sinh ys ds -\- j ds \ . 
Upon integration this equation becomes 

P = Ec/ [sinh 7/ cosh 7/ + 7/ — tanh 7/ (cosh 2 7/ — i) 

2 7 

— 4 sinh 7/ + tanh^ 7/ (sinh 7/ cosh 7/ — 7/) 
+ 4 tanh 7/ (cosh 7/ — i) + 2 7/], 

and when simphfied reduces to 



P=fEJl[3-'^^'^^-t.nh^yl] 



(2) 



as the expression for the total corona loss in watts on an 
open-circuited single conductor. 

As an illustration consider a 60-cycle, three-phase trans- 
mission line with No. 0000 stranded aluminum conductors 
(0.53 inch diameter) placed triangularly 15 feet apart. The 
line constants per mile on a representative single-wire 
circuit which transmits one-third of the total energy, are 
R = 0.463 ohm (includes resistance increase due to skin 
effect and stranding), L = 0.00218 henry, C = 0.0137 
microfarad, and g is negligibly small = o. 

The disruptive critical voltage of this line is given by 
equation (7) of § 72 as 

T7 x/ o^ wQ'5 3 X 2.54 ^ 2 X 15 X 12 

Ecr = 2i.g X 0.85 X log, ^ -= 82 

2 ^' 0.53 

kilovolts to neutral for 5 = i (see also Fig. 93). 

The attenuation and wave-lengths constants per mile are 
respectively 

13 = 0.000563 

and a = 0.00214; 

whence 7 = 0.000563 + 0.002147. 



TRANSMISSION LINES. 251 

If a length of 100 miles (/ = 100) at the open-circuited 
end of the line has a voltage greater than E^r, then 

7/ = 0.0563 + 0.21407, 

sinh 7/ = 0.0550 + 0.21237, 

cosh 7/ = 0.9788 + 0.01207, 

tanh 7/ = 0.0589 + 0.21617, 

3 tan 7/ 

^ — = 3.0308 - 0.02647, 

tanh^ yl = — 0.0432 + 0.02557, 



60 + 25 / a53 

I ^2 X I' 



i^ = 3.9-^-^X/.-Y77^7^-^^-7, 



and 



12.7 

P = (S2y TOO [3 — (3.0308 — 0.02647) 

- (- 0.0432 + 0.02557)] 

= 53,000 watts, or 53 kilowatts. 
Whence the corona loss on the three wires would be 

3 X 53 = 159 kilowatts. 

83. Lightning. — The physical processes^ accompanying 
the establishment of atmospheric differences of potential, 
resultant discharges from which are known as lightning, are 
not well understood. Closely related to the phenomenon 
are two facts established by somewhat recent experiments. 

As the result of the presence in the earth of radioactive 
substances and the characteristics of their decay, the lower 
strata of the atmosphere are partially ionized. The num- 
ber of positive ions per unit volume usually exceeds the 
number of negative ions. This excess seems to disap- 
pear at an elevation of about 10 miles. The resultant posi- 
tive volume electrification estabhshes a positive potential 



252 



TRACTION AND TRANSMISSION. 



in the various strata with respect to the surface of the 
earth. Fig. 103, due to Liebenon, shows the calculated 
potential differences for strata of various altitudes, and is 
based upon experimental evidence. 

Air saturated with water vapor requires the presence of 



•^uu 




















































150 

CO 


















^ 


-^ 


















> 100 


-J 




/ 


/ 






















/ 






















50 


/ 


f 






















/ 


























/ 

























10 20 30 40 

ELEVATION IN THOUSAND FEET 

Fig. 103. 



50 



60 



solid nuclei in order that the vapor may condense to form 
the globules which constitute a cloud. Frequently these 
nuclei consist of dust particles. Kelvin showed that the 
necessity of a nucleus was due to the influence of curvature 
of surface upon the vapor tension, because the greater the 
curvature of a liquid surface the more it tends to evaporate. 
J. J. Thomson showed that electrification would partially 
neutralize the effect of curvature; and C. T. R. Wilson 
showed that ionized air required less supersaturation to 
effect cloud formation than non-ionized air and that nega- 
tive ions were more effective than positive ions. Since 
uncharged globules of a cloud continually move under the 



TRANSMISSION LINES. 253 

influence of the excess of gravitational force above the 
force of air resistance, and since charged globules move as 
the result of an additional force due to the presence of 
the electric field, — positive or negative according to the 
sign of the charge, — it is reasonable to believe that these 
forces contribute towards the establishment of potential 
differences between different parts of a cloud, between 
clouds, and between a cloud and the earth. Under poten- 
tial differences of sufficient magnitude the intervening air 
breaks down accompanied by a discharge. 

The gradual formation of a cloud over a transmission line 
electrostatically induces a charge in the line wires and 
holds it bound. Upon the neutralization of the cloud 
potential by discharge, the energy of the charge on the 
lines is delivered to the line, and tends to dissipate itself 
under conditions prescribed by the constants of the line 
and its environment. Current surges may be set up in the 
line circuit and be superposed upon the normal currents, 
which surges will cease when the energy has been expended 
in heating the conductors, or an arc may be initiated 
between a wire and ground over an insulator or between 
two wires. The subsequent maintenance of the arc will 
be due to energy supplied by the generator. The current 
in an arc to ground is generally intermittent and, if main- 
tained, may set up resonant currents in apparatus con- 
nected with the line, since each piece of apparatus has a 
natural frequency of its own. These resonant currents are 
likely to be accompanied by voltages of magnitude suffi- 
cient to destroy insulation and cause short circuits. 

The energy of the magnetic field associated with a short 
circuit between line wires is delivered to the line when the 
short circuit ceases, and may cause surges similar to those 



254 TRACTION AND TRANSMISSION. 

which result from hghtning. Some writers have therefore 
extended the meaning of the term ^^ hghtning" to include 
such phenomena. 




Fig. 104. 

84. Protection from Lightning. — In order to protect 
apparatus from the high voltages due to lightning it is 
common to insert choke coils, Fig. 104, in series between the 
apparatus terminals and the line wires so that the incoming 
high-voltage wave front may be retarded thereby for a 
short interval of time. On the line side of the choke coil 
is installed a grounded device which conductively connects 
the line with the ground whenever the voltage of the line 
exceeds a predetermined value. This device is termed a 
lightning arrester, and its operation, in connection with the 
choke coil, quickly relieves the line of excessive potentials. 
Some means must be employed, however, to prevent the 
maintenance of a discharge at normal voltage from the line 
to ground over the path rendered conductive by the initial 
discharge under excessive potentials. In nearly all types 
of arresters the circuit from the hne wire to the ground is 



TRANSMISSION LINES. 



255 



normally interrupted by a short dielectric gap which will 
break down under a slight excess over normal voltage. 
The various arresters differ from each other in the means 
employed to suppress the subsequent flow of current at 
normal voltage. In one type this is accomplished by 
separating the spark-gap electrodes by means of a plunger 





Fig. 105. 

solenoid; in another there is an electromagnetic blow-out; 
and in another, for use on alternating circuits, there is a 
series of gaps between electrodes which will not permit 
the maintenance of an arc at normal potentials. 

Two types of lightning arresters have proved particularly 
effective in the protection of station apparatus, namely the 
aluminum-cell and the oxide-film arresters. The former 



256 TRACTION AND TRANSMISSION. 

consists of a series of aluminum cup-shaped electrodes upon 
whose surfaces are formed films of aluminum hydroxide, 
immersed at short distances from each other in a suitable 
electrolyte. The cross section of such an arrester is shown 
at the left in Fig. 105. It is characterized by the conduc- 
tion of minute currents at normal voltage and of very large 
currents, without much elevation of temperature, at volt- 
ages slightly in excess of normal. The number of electrodes 
or cells depends on the operating voltage, each cell with- 
standing permanently about 300 volts. To avoid loss of 
energy under normal potentials these arresters are con- 
nected in series with a horn gap. This practice necessitates 
daily charging of the arrester, as the film dissolves rapidly. 
The oxide-film lightning arrester consists of a series of 
circular sherardized iron electrodes separated by porcelain 
rings, the space between the electrodes being filled with lead 
peroxide, a good conductor, applied under moderate pres- 
sure. When a current is passed through the arrester heat 
is developed at the contacts of the peroxide and metal 
because of the contact resistance, and when the temperature 
reaches about 150 deg. cent, the peroxide at the metal 
surfaces is reduced to a lower oxide, red lead, an insulator; 
consequently obstructing the current flow. The number 
of peroxide layers or cells in the arrester depends upon the 
operating voltage, allowing about 250-400 volts per cell. 
When subjected to an over voltage the insulating films are 
punctured and the discharge takes place through the lead 
peroxide, but the dynamic current following this discharge 
converts the surfaces of the peroxide into the lower non- 
conducting oxide, and thus seals the punctures in the films. 
Commercially, the films are put on initially by dipping the 
plates in a suitable insulating varnish, and after assembly 



TRANSMISSION LINES. 



257 



the passage of a current seals any openings that may exist 
in the varnish film. The arrester is used with a spark gap 
in series, and is depicted at the right of Fig. 105, the metal 
housing being shown removed at one side. The fact that 
this arrester need not be charged extends its use to localities 
where there are no attendants. 

No effective means has been found for the protection of 
a transmission line from a direct stroke of lightning. Such 
strokes usually result in short circuits and shuttered insu- 
lators. The damage is usually confined to one tov/er on 
metal tower lines, but extends over several poles when the 
cross arms and poles are of wood. 

When the stroke is not direct but in the vicinity of the 
line, a common result is a spill-over or arc to ground over 



SELECTIVE 
RELAY 

tt: 



11 n 

OIL 
SWITCH 



Fig. io5. 

an insulator. The maintenance of the arc after the stroke 
by energy from the generator is likely to destroy the insu- 
lator, to set up surges, and to interrupt the service. To 
interrupt such arcs, E. E. F. Creighton has devised a sup- 
pressor, which automatically grounds the affected line at 
the station for a short interval of time, sufficient to allow 
the conducting vapors to escape and the insulator to cool 
off. This time is not so great as to interrupt the service 
because of the slowing down of synchronous apparatus. 



258 TRACTION AND TRANSMISSION. 

The arc ceases because the ground at the station robs it of 
its potential. Fig. 106 is a diagram of the circuit connec- 
tions. The selective relay, which controls the operation of 
the grounding oil switch, is itself controlled by electro- 
static forces on high-voltage lines and by electromagnetic 
forces on moderate-voltage lines. The relay contact is 
normally held open by these balanced forces, but is closed 
when the balance is destroyed. 

Success has been attained in protecting lines by ground 
wires erected above the line and connected with ground at 
every fifth pole or so. The use of such wires has resulted 
in a reduction of 50 per cent in insulator failures. 

PROBLEMS. 

42. Plot a curve showing the resonant frequency of open-circuited trans- 
mission lines of various lengths when connected to impedanceless generating 
units. What length of line corresponds in periodicity with the fifth har- 
monic of a wave whose fundamental frequency is 25 cycles? 

43. Determine the economic voltage to be employed in transmitting 
15,000 kilowatts at 25 cycles to a single substation over a 120-mile three- 
phase aerial transmission line using aluminum conductors. Take the equiv- 
alent annual hours of operation as 4000, the mean annual power factor as 
0.85, the cost of line material as 0.24 dollars per pound, and all other 
factors as suggested in § 71. 

44. Calculate the critical disruptive voltage for the transmission line 
discussed in § 81 in a region where the atmospheric pressure is 70 cm. of 
mercury and the temperature averages 20 deg. cent. 

45. Determine the line constants per mile per phase at 15° C. of a three- 
phase 60-cycle aerial power transmission line using solid hard-drawn copper 
conductors 0.8 inch in diameter spaced triangularly 6 feet apart. 

46. Calculate the voltage and current at the generator end of the line, 
the efficiency of transmission, the voltage regulation, and the charging 
current of the transmission line of § 81 when the frequency is 25 cycles, all 
other conditions remaining unaltered. 

47. Plot a curve showing the corona loss per mile of the transmission 
line of § 81 for various operating voltages up to 200 eflFectlve kilovolts be- 
tween conductors. Take the density factor as unity. 



POWER STATIONS. 



259 



CHAPTER X. 
POWER STATIONS. 



85. Station Load Curves. — The proper design of a power 
station depends to a large extent upon the characteristics 
of its output. A curve with ordinates representing the 
output of a station in kilowatts and with corresponding 



'(■U 










A 








/ 


\ 






CO 

t- 

<30 










/ 


\ 








/ 


\ 












/ 


\ 


\ 




/ 


/ 


\ 






ii. 
020 

CO 



z 








/ 




V 




/ 




\ 












/ 














\ 




< 

CO 

10 


N, 




/ 
















\ 


^_ 


\ 




J 















































12 


2 


4 


6 


8 10 12 2 4 


6 


8 


10 12 


A.M. 








TIME IN HOURS 






P.M 



Fig. 107. 

abscissae representing the time of day is termed a load 
curve of the station. Fig. 107 represents a typical load 
curve for a power station supplying energy for traction 
purposes. It is characterized by two peaks, which occur 
at about 8.30 in the morning and 6.00 in the evening 
respectively, and which last for two or three hours, and by 



26o TRACTION AND TRANSMISSION. 

a very low value during the early morning hours. The 
peaks are due to the demands of traffic in carrying pas- 
sengers to business in the morning and returning them 
to their residences at night. The maximum value of the 
peak at the power station is less than the sum of the peaks 
at the different substations; because the latter occur at 
different times, that is, because of the diversity factor. In 
the morning, the peaks at the substations in the residential 
districts occur prior to those in the business and manu- 
facturing districts, while the reverse is true in the evening. 
Furthermore, the average duration of the power-station 
peaks is greater than characterizes the substation peaks 
for the same reason. The ordinates of the load curve are 
greater in winter than in summer because of the necessity 
for heating and lighting the cars, and often because of the 
presence and removal of snow. The energy required for 
heating may be 20 per cent of that required for car propul- 
sion. The shape of the load curve is likely to be entirely 
different on Sundays and holidays from its shape on week 
days and may be materially modified by the maintenance of 
seasonal amusement or recreation resorts. Instantaneous 
fluctuations in the power output, not shown in the load 
curve and due to the abnormal currents necessary in the 
starting of trains, are always present. With few cars in 
operation the relative magnitude of these fluctuations is 
greater than when there are many. The amount of fluc- 
tuation can be determined with sufficient exactitude from 
the curve of Fig. 58, which shows the dependence of the 
ratio of maximum to average current upon the number 
of cars in operation. 

The power-station load curve for a proposed installation 
can be predetermined with considerable accuracy from the 



POWER STATIONS. 261 

train-sheet, § 52, of the tentative service to be maintained 
and the curves of power input to the car, § 43, for different 
times. The ordinate of a point on the power-station load 
curve for a given instant is equal to the sum of the inputs 
to all cars in operation at that instant, divided by the 
product of the efficiencies of transmission, of conversion, 
and of distribution, which product usually ranges from 
70 per cent to 75 per cent. With urban systems, where 
congestion of street traffic constantly interferes with regu- 
larity of schedules, this method is inapplicable. In such 
cases a fair estimate of the power-station output in kilo- 
watts at any instant is, however, numerically one-half the 
rated horsepower of all the motors on cars in service at that 
instant. This method of estimation is based upon the fact 
that the continuous current capacity of a railway motor 
is about one-half its capacity when nominally rated in ac- 
cordance with the Standardization Rules of A.I.E.E. The 
average power supplied to a certain number of cars is there- 
fore one-half the rated horsepower of the corresponding 
motors, and with an efficiency of 75 per cent in transmission 
from the power station to the cars a kilowatt at the station 
corresponds to a horsepower at the car. 

86. Selection of Generators. — For stations of small 
capacity supplying energy for short roads it is often eco- 
nomical to use 2 200- volt generators, as the cost of wiring is 
less than for lower voltages and the cost of insulation is less 
than for higher voltages. Furthermore, this being a stand- 
ard voltage for lighting generators, there is a complete 
line of these generators available. For systems where the 
economic voltage for transmission, calculated under the 
assumed use of step-up transformers, is of the order 20 

kilovolts, standard generators wound for 12 kilovolts and 
18 



262 TRACTION AND TRANSMISSION. 

connected directly to the transmission line will generally 
prove more economical. For transmitting large amounts 
of power at higher voltages step-up transformers must be 
used while the generator voltage should conform with stand- 
ards such as 6.6 or ii kilovolts. 

The size of a unit, including generator and prime mover, 
should be such as to entail a minimum annual charge against 
it, arising from its cost and operation. To reduce the 
relative losses in a unit it should be operated as nearly as 
possible at that load which gives a maximum efficiency. 
Because the designed operating efficiency is generally great- 
est at about rated load and, because of the characteristics 
of the load curve, the losses would be least with units of 
minimum rated capacity. The efficient use of such small 
units, however, would necessitate frequent starting and 
stopping of the different units corresponding to the fluctu- 
ations of load, and this would require a large force of 
attendants. Furthermore, the cost, the deficiency, and the 
required floor space per kilowatt is greater for small units 
than for large ones, and therefore the proper selection is, 
by nature, a compromise. 

Very small stations are generally located upon cheap 
land and space economy is of no great importance, whereas 
the number of attendants must be reduced to a minimum. 
Furthermore, the cost per kilowatt varies so greatly with 
the capacity of small units that, if capital is limited, it may 
be necessary to install but a single unit. For the sake of 
reliability of service, however, it is undesirable to use less 
than two units. 

For the average station of moderate capacity four units, 
one of which serves as a reserve unit, to be used in case of 
failure of another, will generally prove most economical. 



POWER STATIONS. 263 

The relative values of the early morning and noonday 
loads, which endure for protracted periods, may, however, 
make it desirable to use a larger number of units so as to 
operate at good efficiency during these hours. 

Very large stations have been installed in the past with 
the number of units prescribed by the maximum capacity 
available. Steam-turbine units are now constructed which 
have a rated capacity of over 45,000 kilowatts. 

According to the standardization rules of A.I.E.E. gen- 
erators should be able to carry a 25 per cent overload for 
two hours. If a railway power station were to be equipped 
with five units, each of rated capacity equal to one-fifth 
the maximum station load, then in case one should fail 
the whole load could safely be carried by the remaining 
four. This is possible because the fifth unit is seldom in 
service for more than two hours during the peak loads. A 
reserve unit may thus be dispensed with. If the power 
factor of the load on the generators be less than unity, the 
overload capacity may not be sufficient as a substitute for 
the reserve unit. 

87. Types of Prime Movers. — The types of prime 
movers at present available for electric power stations 
are steam engines, internal combustion engines, and water 
wheels. As a rule that type should be employed which 
will result in a minimum average cost of reliably delivering 
a kilowatt-hour of energy. To make an equable com- 
parison the point of delivery should be the same in all 
cases. This will generally require for hydraulic plants that 
a part or the whole of the expense of the transmission sys- 
tem shall be considered as chargeable to the power station. 
If the financial hazard associated with the undertaking be 
large or if capital be limited, it may be necessary to reduce 



264 TRACTION AND TRANSMISSION. 

the first cost, the plant thereafter being burdened with an 
excess cost of energy production. 

Internal combustion engines burning gas or liquid fuel 
in their cylinders have a high thermodynamic efficiency. 
The high pressures developed require heavy construction, 
the high temperatures require cooling systems, and the 
intermittent release of energy requires heavy flywheels. 
They therefore cost more than other forms of prime movers, 
and depreciate in value faster. Furthermore, gas engines 
have a very limited overload capacity. Reliability in their 
operation has not been sufficiently established to warrant 
the recommendation of their adoption as a sole source of 
power in a station for supplying energy for railways. Yet 
the Milwaukee and Northern road as well as the Warren 
and Jamestown road are operated solely from generators 
driven by gas engines. 

88. Power Station Costs. — The annual cost of operat- 
ing a station is conveniently divided into two parts, namely, 
fixed charges which do not vary with or depend upon the 
output of the station after it is built and equipped, and 
operating expenses which vary with the output. The 
fixed charges usually comprise interest, taxes, insurance, 
rental, depreciation, and obsolescence. Sometimes there is 
apportioned to the power station a part of the annual 
administration costs, including office rentals, salaries, and 
legal expenses. The operating expenses comprise labor or 
attendance, repairs and maintenance, fuel, water, oil, waste, 
and other supplies. 



STEAM STATIONS. 265 

STEAM STATIONS. 

89. Engines and Turbines. — Steam-driven prime movers 
may consist of reciprocating engines or turbines, operated 
with or wdthout exhaust steam condensers. The former are 
usually either simple or compound and are sometimes clas- 
sified as high-speed or low-speed, although there is no 
sharp dividing line in this respect. A speed of 150 revolu- 
tions per minute may be assumed as the usual line of 
division. The proper selection of a prime mover of this 
type is based upon the first cost of the prime mover and of 
the rest of the equipment entailed by its use, as well as 
upon the expenses of maintenance and operation. Data 
concerning steam prime movers generally include pounds 
of steam consumed per indicated horsepower-hour or per 
kilowatt-hour of output, initial and back pressures of 
the steam, and the mechanical efficiency of the mover. 
The steam consumption and efficiency vary with the load, 
as doe^ the efficiency of a generator. With assumed con- 
ditions as to pressures and load, the pounds of steam per 
kilowatt-hour of generator output is to be found by dividing 
the pounds of steam consumed per indicated horsepower- 
hour by 0.746 times the product of the generator and 
prime-mover efficiencies. The steam consumption of re- 
ciprocating engines increases somewhat with use, whereas 
that of turbines remains fairly constant. The steam con- 
sumption of Curtis turbines decreases about one percent for 
each increment of 10 pounds in gauge pressure and one 
pound per kilowatt-hour per inch of vacuum. 

At a given pressure, steam having the minimum tempera- 
ture consistent with its remaining in the form of a vapor 
is termed saturated steam, and a reduction of its tempera- 



266 TRACTION AND TRANSMISSION. 

ture causes condensation. If saturated steam be removed 
from contact with water, its temperature may be raised 
above that of the water from which it was produced. It 
then acts like an imperfect gas and is termed superheated 
steam. The rise of temperature in degrees Fahrenheit is 
a measure of the amount of superheat. If steam rises 
from a surface of water faster than about three feet per 
second, it carries water with it in the form of spray, and 
when fine spray is once formed in steam it does not readily 
settle. The resultant mixed steam is termed wet steam. 
Superheated steam, if homogeneous, cannot be wet, be- 
cause water particles would of necessity be evaporated 
under the influence of heat derived from the surrounding 
steam. 

The cyclical changes in the temperature of cylinder walls, 
accompanying the operation of reciprocating engines, causes 
cylinder condensation losses of heat when it is fed with 
saturated steam. Such losses are seldom less than lo per 
cent and often amount to 40 per cent of the supplied -energy , 
and may be materially reduced by the use of superheated 
steam. The presence of moisture in the steam passing 
through a turbine occasions a wear of the turbine blades 
as the result of impact. It is therefore desirable to supply 
superheated steam to reciprocating engines on the ground 
of economy and to turbines on the ground of maintenance. 
A device used to elevate the temperature of steam above 
its saturation temperature is termed a superheater and 
may consist of a set of tubes connected in the steam line 
and subjected to the heat from the fire of the main boiler 
or from an auxiliary source. 

The data contained in the following table give an idea 
of what may be expected as to the performance of these 



STEAM STATIONS. 



267 



types of prime movers. The efficiency of reciprocating 
engines and of generators has been assmiied as 92 per cent 
and 97 per cent respectively. 



STEAM COXSUMPTIOX. 



Type of engine. 


Pounds of steam 
per K.W.H. 


Saturated Stea:m: 

Simple noncondensing 


55 
35 
33 
27 
20 

14 
15 


Compound noncondensing 


Simple condensing 

Compound condensing 

Turbines 


Superheated Steam: 

Compound condensing 

Turbines 





90. Condensers. — Consider a simple engine run so that 
the steam after expansion exhausts into the atmosphere; 
that is, run noncondensing. The effective force per unit 
area of piston, available at any instant for performing 
work, is the difference between the pressure of the steam 
on one of its surfaces and the back pressure exerted by the 
atmosphere at that instant on the other surface. Since the 
mean effective value of the former may be of the order 
>,o lb. in.- and the latter is 14.7 lb. in.-, a reduction of the 
latter to 1.7 would theoretically increase the power out- 
put 13 50 or 26 per cent. An enclosed device which is 
adapted to receive the exhaust steam, lower its tempera- 
ture, and thereby condense it, is termed a condenser. Its 
use materially reduces the back pressure because steam, 
after condensation, occupies an insignificant portion (ttVo) 
of the space filled by it prior to condensation. In order 
to cool and condense the steam it must be deprived of 



268 TRACTION AND TRANSMISSION. 

some of the heat associated with it. This may be done 
by passing it along one surface of a thin metal which is 
kept cool by water circulated in contact with the other 
surface or by mixing the steam with a spray of cooling 
water. A device using the first method is termed a sur- 
face condenser J and one using the latter is termed a jet 
condenser. The condensing water used with the jet con- 
denser is variously termed, as injection, cooling, or circulat- 
ing water. To maintain the condenser in operation the 
condensed water, which has collected in a hot well, must be 
removed by a wet-vacuum pump, which may also serve to 
remove the air which is invariably present as the result of 
leakage, or absorption in the injection water. To main- 
tain a high vacuum an additional dry-vacuum pump is often 
used for removing the air. 

The amount of cooling water required per pound of 
condensed steam depends upon the vacuum and upon the 
initial and final temperatures of the cooling water. 

Let X = total heat of the exhaust steam above 32° F., 
Tq = initial temperature of the cooling water, 
rp _ { temperature of the condensed steam (surface) , 

} temperature of the discharge water (jet), 
T2 = temperature of the discharge water. 

Then the weight of cooling water, W, necessary to con- 
dense one pound of saturated steam, is 

W - _ 'Z.^ pounds. 
I2 — J^ 

Surface condensers cost more than jet condensers, but 
permit the use of the condensed steam as feed water for 
the boilers after any oil, which became mixed with it in 
the engine, has been removed from it. They are there- 



STEAM STATIONS. 



269 



fore adapted for use where there is a hmited supply of 
suitable feed water but a superabundance of cooling water, 
such as results from a location near salt waterways. 
When the supply of cooling water is limited the use of 
cooling ponds or cooling towers permits of the repeated use 
of the same water, but these arrangements are expensive. 

The advisability of installing condensers depends upon 
whether the annual saving of energy is greater or less than 
the annual expense entailed 
by their cost, maintenance, 
and operation. 

A jet condenser is shown 
in Fig. 108 with parts cut 
away so as to indicate the in- 
terior construction. The ex- 
haust steam enters through 
the large pipe at the left and 
the cooling water through 
the large pipe at the right. 
The latter is sprayed through 
the valve in the center, 
mixes with the steam, con- 
denses it, and both fall into 
the pipe below. The air- 
pump is connected with the 
small pipe at the left. With 
the surface condenser shown in Fig. 109, the cooling water 
is passed through the interior of the small tubes and ab- 
stracts heat from the exhaust steam, which surrounds the 
tubes, thereby condensing it. The circulating pump to 
the right and the vacuum pump to the left are operated 
by an intermediate auxiliary engine. 




Fig. 108. 



270 TRACTION AND TRANSMISSION. 

91. Boilers. — An essential element in a steam plant is 
the boiler equipment, and its size and cost depend upon 
the amount of steam which is to be supplied to the prime 
movers and to the auxiliaries. A typical form of boiler for 
use in power stations is shown in Fig. no, wherein the water 
to be heated circulates as the result of localized tempera- 
ture differences, moving to the right in the cylindrical 




Fig. 109. 

drum at the top, and to the left in the water tubes ^ which are 
enveloped in the hot gases resulting from the combustion 
of the fuel. These gases ultimately pass through the 
damper-zoniroWed opening near the top of the right-hand 
enclosing brick wall, and through a breeching to the chimney 
or stack. Steam is generated and confined under pressure 
in the upper part of the drum, and is fed through the nozzle 
on top to a header, whence it is conducted direct to the prime 
mover. The capacity of a boiler is rated in horsepower 



STEAM STATIONSc 



271 



and the builder^ s rating is based upon a heating surface of 
10 to 12 square feet per horsepower. A boiler of one 
horsepower capacity is considered to be capable of allowing 
an evaporation of 34.5 pounds per hour of water at 212° F. 
into steam at atmospheric pressure, and to have an over- 
load capacity of 33^ per cent. If the temperature, /, of 
the feed water be less than 212°, the steam be x part dry^ 
or the steam be superheated // F., the delivery of 34.5 




Fig. no. 



pounds of steam per hour under such conditions will re- 
quire a boiler of more than unit capacity, and to deliver Q 
pounds of steam per hour the horsepower capacity of the 
boiler should be 



Q f xr + q + Ct, - t + 32 



] horsepower, 



where r = latent heat of evaporation at the resultant 
pressure, 
q = heat in liquid at this pressure, and 
C = mean specific heat of the superheated steam. 



272 TRACTION AND TRANSMISSION. 

The values of the various constants may be found in 
Engineering handbooks. 

The steam consumed in operating auxiharies such as 
feed pumpSj vacuum pumps, and circulating pumps, ranges 
from 6 per cent to 15 per cent of that taken by the prime 
movers. Available boilers are limited in capacity to about 
2250 horsepower, and it is common to install smaller ones 
in batteries of two or more. 

92. Feed-water Heaters. — It is undesirable to pump 
cold water into a hot boiler because of excessive stresses 
which may result from wide differences in the temperature 
of adjacent parts of the metal of the boiler. Furthermore, 
there is a saving of about one per cent in fuel for every 
II degrees elevation in the temperature of the feed water, 
provided such elevation is produced by heat that would 
otherwise be lost. The temperature of the feed water may 
be raised by heat taken from the exhaust steam through 
the aid of a vacuum heater or an atmospheric heater ^ and by 
heat from the hot flue gases, using an economizer, 

93. Chimneys or Stacks. — A chimney serves two pur- 
poseS; namely, to carry off the obnoxious gases resulting 
from combustion, and to produce a draft which will give a 
sufficient supply of oxygen for combustion. The former 
requires an adequate cross section and the latter an ade- 
quate height of chimney. Experience shows that the draft 
pressure, measured in inches of water as compared with 
atmospheric pressure, should be from 0.5 to 1.5 inches, de- 
pending upon the character and size of the fuel to be used, 
and upon the quantity to be burned per square foot of grate 
surface. Heights above the grate, which have given satis- 
factory results in practice with plants of moderate capacity 
employing different fuels, are given in the following table: 



STEAM STATIONS. 

HEIGHTS OF CHIMNEYS. 



273 



Fuel. 


Height in feet. 


Free- burning bituminous 

Anthracite, large sizes 

Slow-burning bituminous 

Anthracite buckwheat 

Anthracite slack 


80 
100 
120 
150 
175 





The ascending gases in a chimney are retarded by fric- 
tion in the vicinity of the walls, and the equivalent cross 
section ^ of a round chimney is therefore generally taken 
as that corresponding to a diameter four inches less than 
the real internal diameter of the chimney. Assuming a 
coal consumption of five pounds per horsepower-hour, 
a chimney of height h feet, properly to carry off the gases 
from boilers of P horsepower, should have an equivalent 
cross section of 

. o.^P , ^ 

A = ^- square feet. 

y/h 

Chimneys are constructed of steel, reenforced concrete, 
or masonry. Steel chimneys weigh less, cost less, require 
less space, expose less surface to the wind than other forms, 
and are more efficient because they are air-tight. They, 
however, depreciate more rapidly because of rust and be- 
cause of the corrosive influence of the flue gases. 

Sometimes short chimneys are used in connection with 
mechanical draft apparatus, consisting of either an exhaust 
fan in the smoke flue or a mechanical or steam-jet blower 
underneath the grate bars. An induced draft is produced 
by the former and a forced draft by the latter. The advis- 
ability of installing mechanical draft apparatus is depend- 
ent upon the results of an economical comparison with 



274 



TRACTION AND TRANSMISSION. 



the saving resulting from the lessened necessary height of 
chimney. 

94. Buildings. — Power-station buildings may be con- 
structed of wood, brick, reenforced concrete, or stone. 
Wood is used only for very small stations and stone only 
for elaborate stations. If a single building is used for 
housing the boiler plant as well as the generating plant, the 



^3 

u 
ol 

lli 2 

hi 
L- 

U 
DC 
< 

o-l 



10 20 30 40 50 60 70 80 90 100 
THOUSANDS OF KILOWATTS 

Fig. III. 

two should be separated by a brick wall with no openings 
in it which will allow dirt to pass through from the boiler 
room to the engine room. The boilers and the units which 
are supplied with steam from them should be on opposite 
sides of the dividing wall and so placed as to reduce the 
length of steam piping to a minimum. The height of both 
rooms should be ample, to permit the use of lifting machinery 
and the replacing and repairing of boilers. The building 
should be well Hghted, well ventilated, of fire-proof con- 
struction, and arranged with a view to extension in case of 
growth of demanded output. 























\.. 




















\ 




















.\ 


\- 




















c 


9 




X 















• 


c 







X 

B — 
























• PA 

cu 


^SONS ■ 
RXIS TL 


URBINI 
RBINES 


S. 














X RE 


:iPROcy 


TING E 


^IGINES 









STEAM STATIONS. 275 

The floor space required for turbines is materially less 
than that for reciprocating engines of the same capacity and 
the foundations can be much lighter. Where the cost of 
land is great a considerable saving may be effected by placing 
turbines on a floor above the boiler room. The station is 
then termed a double-deck station. The space required for 
passageway around units is greater per kilowatt for small 
units than for large ones. The curve of Fig. 1 1 1 is based upon 
existing plants, and shows the average floor space allowed 
per rated kilowatt in terms of the total capacity of a plant. 

95. Arrangement of Apparatus. — It is customary to 
arrange the apparatus in a steam-power station so that the 
path of energy is as short as possible. The coal is there- 
fore received and delivered to the boilers at one end of the 
station and the electrical energy is delivered to the line 
from the generators at the other end. Figs. 112 and 113 
show an elevation and floor plan of the Winona Interurban 
Railway Power House which has a capacity of 1200 K.W. 
The output is supplied at 33,000 volts from two banks of 
three transformers, each of 200 K.W. capacity and stepping 
the voltage up from 2300 volts. There are two 600-K.W. 
25-cycle, 2300- volt generators, each directly connected to a 
cross-compound engine guaranteed to have a full-load 
steam consumption not to exceed 14. i pounds per indicated 
horsepower-hour at 140 pounds pressure and 26 inches of 
vacuum. Each engine is supplied with a jet condenser. 
Steam is supplied by four boilers, arranged in batteries of 
two each, there being 3000 square feet of heating surface 
provided in each unit. It will be noted that a transformer- 
converter substation, for supplying 600-volt direct current 
to the distribution circuits in the immediate vicinity is 
housed under the same roof. 



276 



TRACTION AND TRANSMISSION. 




STEAM STATIONS. 



277 



Fig. 114 shows a cross section of the Port Morris Power 
House of the New York Central Railroad, which is equipped 
with Curtis steam-turbine units and surface condensers. 



COAL SHED 



COAL TRACK 



77M.W/////////,7777777^/////////////////////^^^ 



FEED PUMPS. SERVICE PUMPS. 




FUTURE 
VA .rr. 0"- SWITCH _ 



OIL SWITCH 



OIL SWITCH 



^^ 



^^^3 




Fig. 113. 



The very complete system of labor-saving apparatus for 
conveying coal and removing ashes and its method of 
operation is clearly shown. 
19 



278 



TRACTION AND TRANSMISSION. 




STEAM STATIONS. 



279 



POWER-PLANT COSTS PER KILOWATT. 



1 . Real estate 

2. Excavation 

3. Foundations, reciprocating engines 

4. Foundations, turbines 

5. Iron and steel structure 

6. Building (roof and main floor) 

7. Galleries, floors, and platforms 

8. Tunnels, intake and discharge 

9. Ash storage pocket 

10. Coal hoisting tower 

1 1 . Cranes 

12. Coal and ash conveyors 

13. Ash cars, locomotives, and tracks 

14. Coal and ash chutes 

15. Water meters, storage tanks, and mains. . . . 

16. Stacks 

17. Boilers 

18. Boiler setting 

19. Stokers 

20. Economizers 

21. Flues, dampers, and regulators. 

22. Forced draft blowers, air ducts 

23. Boiler, feed, and other pumps 

24. Feed- water heaters 

25. Piping, traps, and separators 

26. Pipe covering 

27. Valves 

28 Main engines, reciprocating 

29. Exciter engines, reciprocating 

30. Condensers, barometric or jet 

31. Condensers, surface 

32. Electric generators 

SS- Exciters 

34. Steam-turbine units, complete 

35. Converters, transformers, blowers 

36. Switchboards, complete 

S7. Wiring for lights, motors, etc 

38. Oiling system 

39. Compressed air system and other small aux- 

iliaries 

40. Painting, labor, etc 

41 . Extras 

42. Engineering expenses and inspection 



Min. 



^3.00 

■75 
2 .00 

•50 
8.00 
8.00 
1.50 
1 .40 

.70 
1 . 20 

.40 
2 .00 

15 
.40 

•50 
1-25 
9-50 
1-25 
1.30 
1.30 



.60 

1-25 
.40 
. 20 

3.00 

.60 

.60 

22 .00 

.40 

1 .00 

6.00 

16.00 

.60 

22 .00 

.60 

3.00 
. 20 
•15 

. 20 

1-25 

2 .00 

4. CD 



Max. 



$7- 00 
1-25 
3.00 

•75 

10.00 

10.00 

2.50 

2.80 

I -50 

2 .00 

.60 

2-75 

•30 

1 .00 

1 .00 

2 .00 

11.50 

1-75 
2 . 20 
2.25 

.90 
1-65 

-75 

•35 

5.00 

1 .00 

1 .00 

30.00 

.70 
2.50 

7^50 

22 .00 

.80 

32.00 

1 .00 

3-90 

•30 

•35 

•30 
1-75 
2 .00 
6.00 



28o 



TRACTION AND TRANSMISSION. 



96. Cost of Steam Stations. — The table on the preced- 
ing page, due to H. G. Stott, includes the approximate 
cost per kilowatt of the various elements entering into the 
cost of a steam plant. A fair average cost per kilowatt is 
^100 for plants using reciprocating engines and $&o for 
those using steam-turbine units. 

97. Operating Expenses. — Data concerning twenty- 
three stations of moderate capacity, using mostly bitu- 
minous coal ranging in price from ^2.75 to ^5 per gross 
ton, and all operated condensing, has been published re- 
cently by E. F. Tweedy. Fig. 115 shows the operating costs 



cr O 

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Xx 



50 

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o 
o 



3 




X 




















































RECIPROCATING STEAM ENGINES. 
©STEAM TURBINES. 

>< MIXED EQUIPMENT-ENGINES & TURBINES. 
EQUATION OF HYPERBOLIC CURVE 
y- 1 4. 900,000 






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1 2 3 4 6 6 7 8 

MILLIONS OFKILOWATT HOURS GENERATED PER YEAR. 

Fig. 115. 



per kilowatt-hour in terms of total annual outputs. The 
highest load factor based upon rated capacity was 0.23, 
the lowest o.ii, and the average 0.17. The coal consumed 
per kilowatt-hour ranged from a little over 3 pounds for 
the larger plants to about 5 pounds for the smaller ones. 
The station rating in kilowatts per man employed in 
operating the station, ranged from about 100 K.W. for the 



HYDRAULIC STATIONS. 



281 



smallest stations to 250 K.W. for the largest. Fig. 116 
shows the percentage distribution of operating costs among 
fuel, labor, and miscellaneoias items. 



o 
o 



100 
90 
80 
70 
60 
50 
40 
30 
20 
10 





















MISCELLANEO^US COSTS 

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12 3 4 5 6 7 8 

MILLIONS OF KILOWATT HOURS GENERATED PER YEAR. 

Fig. 116. 



HYDRAULIC STATIONS. 

98. Turbines. — In procuring mechanical energy from 
water powder two classes of turbines or water wheels may 
be utihzed in conformity with American practice; namely, 
the reaction turbine and the impulse wheel. 

The reaction or pressure turbine of the mixed-floiv type 
is appUcable for low and moderate heads, say up to 150 feet, 
although this type has been used for heads up to 600 feet. 
It consists of a rotating wheel or runner carrying vanes or 
buckets to which water under pressure is delivered radially 
inward by means of stationary guide vanes surrounding the 
wheel, and from which the water is discharged partially 
in an axial and partially in a radial direction. Torque 
is developed by reaction, due to changing the direction of 
water flow. 



282 TRACTION AND TRANSMISSION. 

As the buckets and wheel passages are always completely 
filled with water, it is not necessary to mount the turbine 
at the level of the discharged or tail water in order to realize 
the total head, if an air-tight draft tube leading from the 
wheel outlet down somewhat below the level of tail water 
be provided; for the falling water in the draft tube from 
the turbine creates a vacuum that is effective in sucking 
the water through the turbine, and which is equivalent to 
increasing the pressure of the inflowing water. Reaction 
turbines may be placed at any level up to about 20 feet 
above the tail race without loss of head. 

The power developed by a turbine under a given head 
is regulated by varying the amount of water admitted to 
the runner by means of gates. There are various types of 
gates, including the so-called cylinder^ register, and wicket 
gates, the last being the most used. In this type the guide 
vanes are pivoted so that all may simultaneously approach 
or recede from their neighbors by the rotation of a single 
regulating shaft. 

In order to neutralize the end thrust due to the axial 
pressure of the water, as well as to secure higher speeds 
under a given head, it is common to place two turbine 
runners — of correspondingly reduced diameter for the 
same total power output — on a single shaft. Sometimes 
four and even six runners are coupled together to constitute 
a single unit. Fig. 117 shows a- 9000 horsepower AUis- 
Chalmers horizontal twin turbine with the runners dis- 
mantled. The water enters through the wicket gates at 
the ends and within the bearings, passes through the wheels, 
and emerges at the bottom. 

Impulse wheels, suitable for heads above 150 feet, com- 
prise a number of buckets into which water is directed 



HYDRAULIC STATIONS. 



283 




284 



TRACTION AND TRANSMISSION. 



through one or more nozzles at a velocity equal to V2 gH 
feet per second, where g is the acceleration due to gravity 
= 32 ft. per sec. per sec, and H is the head or height of 
water in feet. Each bucket forms two cups divided by a 
central ridge which separates the impinging water into two 
parts, each part being deflected backward to one side of the 




Fig. 118. 

wheel by the bucket. The effective head is that from the 
level of headwater to the nozzle, the head from the latter 
to the tailwater being lost; consequently the impulse wheel 
should be placed as low as possible. The flow of water is 
regulated by needle valves or by deflecting the nozzle. 
Fig. 118 shows a twin Pelton water wheel with its ^^ hy- 
draulic relay " governor. 

Governors are used on both types of water turbines for 
automatically effecting the opening and closing of the regu- 



HYDRAULIC STATIONS. 285 

lating gates or for deflecting the jet from the buckets of im- 
pulse wheels. As the force required for this purpose is very 
large, it is evident that the centrifugal ball governor cannot 
directly control the gate opening, but must do so through 
the intervention of a relay. Two general types of relay are 
used : mechanical relays, which derive power for their opera- 
tion from the water wheel by means of gears, pulleys, or 
other mechanical devices, and hydraulic relays, which are 
operated either by the pressure of water taken from the 
'^ penstock '' or other source, or by oil supplied under high 
pressure from a reservoir. 

Turbines or water wheels are ordinarily direct-connected 
to the electric generators, but may be either geared or 
belted thereto, there being one prime mover for each gen- 
erator, and one or more additional turbines for the exciter 
units. Four generator units is considered the minimum 
number allowable for the attainment of a reasonable de- 
gree of insurance against shut-down. 

Having determined the number and size of the electric 
generating units from a study of the load curves on the 
power station, the size of the prime mover in horsepower is 
found by dividing the kilowatt rating of the generator by 
C.746 times the efficiencies of the generator and mover. The 
efficiency of large generators at full load may be taken as 
between 93 and 97 per cent. The efficiency of turbines 
and water wheels is conventionally taken as 80 per cent, 
although efficiencies as high as 86 per cent have been 
attained. Some of the turbines of a hydroelectric power 
house should have a high efficiency at low-gate opening and 
others should have their greatest efficiency at full-gate, so 
as to realize a fairly high all-day plant efficiency under 
widely varying loads. Representative efficiency curves of 



286 



TRACTION AND TRANSMISSION. 



two modern reaction turbines at various gate openings are 
shown in Fig. 119. 

The power developed by a turbine or impulse wheel 
depends upon the quantity of water passing through it in 



lUU 




































>-80 


z 














/ 


^ 

y 


^ 




"^ 



li- 
[i]60 

H 

z 






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/ 


/ 


/ 










LiJ 

^40 




/ 


/ 




























20 



















1/4 V2 3/4 

GATE OPENING 
Fig. iig. 



FULL 



unit time, upon the available head of water, and upon the 
turbine efficiency, e, and is 

„ 62.AqeH QeH . 
P = — =^-^ — = - — - horsepower, 
550 8.81 

where q is the discharge in cubic feet per second and which 
may be expressed empirically as 

q = KU" Vff , 

wherein D is the diameter of the runner in feet, and K is 
an experimental constant of discharge dependent upon the 
design of the turbine. Therefore 



8.81 



horsepower, 



HYDRAULIC STATIONS. 287 

whence the proper wheel diameter for a given head is 

D = \/^feet. (x) 

The values of K vary widely among the different designs of 
various manufacturers, but most values thereof lie between 
2.3 and 3.5 for reaction turbines, and between 0.015 ^^d 
0.024 for impulse wheels. 

For a given turbine the speed of the runner varies with 
the square root of the head. Let r be the r^tiio of the 
peripheral velocity of the buckets to the theoretical velocity 
that water would acquire in falling freely a hei^^ht equal to 
the head of water. Then the speed of the wheel in revolu- 
tions per minute is 

The values of r range from 0.65 to 0.93 with different 
designs of reaction turbines and between 0.43 to 0.51 with 
impulse wheels. Having determined the turbine speed for 
a given head of water, the multipolarity of the alternators 
for the generation of electromotive forces of definite fre- 
quency becomes known. 

As an illustration of the foregoing, determine the proper 
number of poles for a 2000 K.W., 60-cycle, three-phase 
alternator which is to be driven by a Pelton water wheel 
on a head of 970 feet, the constants of the wheel being 
K, = 0.019, r = 0.505, and e = 0.83. Taking the alternator 
efficiency as 92 per cent, the rating of the prime mover is 

= 3500 horsepower and the diameter of 

0.746 X 0.83 X 0.92 



the water wheel is y ^ =8.0 feet. Therefore 



0.019 (970)^ O 84 



288 TRACTION AND TRANSMISSION. 

its speed is ^|^ 0-505 ^^970 = 300 revolutions per minute. 
At this speed there must be 24 poles for the production of 
60-cycle currents. 

99. Water-power Development. — In any hydraulic de- 
velopment the water must be conducted from some source 
to the wheels by means of a head-race, and discharged 
from the turbines into the tail-race at a lower level. Two 
general types of water-power development are discernible 
which usually characterize respectively low-head and high- 
head developments; namely, (i) where the entire head 
is utilized at the dam, the power station being located at 
one end thereof; (2) where long pipe lines ^ canals, ov flumes 
are required to transfer the water from the intake at the 
headworks to the station, this distance being only suffi- 
ciently long to secure for a given amount of water a head 
which will enable the generation of the required power. 

(i) The object of a dam is to concentrate the fall of a 
stream so that the water power becomes available by the 
elevation of the water surface. That portion of a dam 
over which excess water pours is called the spillway, and 
this must be sufficiencly long to allow escape of the water 
in times of heavy flood without undue rise in level of the 
water in the reservoir above the dam. It is essential that 
the dam have a solid foundation, that it be stable against 
overturning and be water-tight, and that it be so con- 
structed as to prevent washing out of the river bed and 
banks below it and erosion of the dam itself. Dams may 
be constructed of timber, masonry, or reenforced concrete. 
They must be equipped with drain or sluice gates for the 
purpose of draining the reservoir above them as well as for 
assisting in the discharge of water during the heaviest 
floods. The surface of the reservoir may be raised at 



HYDRAULIC STATIONS. 



289 



times by means oiflashboards, which collapse automatically 
upon excessive rise of water. 

A plan of a typical low-head hydraulic development is 
illustrated in Fig. 120, which shows the Johnsonville de- 
velopment of the Schenectady Power Company. This dam 
causes the flooding of 850 acres, thereby giving a storage 




DEFLECTING WALL 




SPILLWAY 530 FT. 



SLUICE CATL3 



Fig. 120. 

capacity or pondage of about 350 million cubic feet. Fig. 
121 shows the power house and sluice-gate masonry of this 
development, looking upstream. 

The power furnished by a given stream may be increased 
by a suitable reservoir, for the water impounded during the 
rainy seasons may be partially drawn off during time of low 
water. The water available for pondage is limited, how- 
ever, since the level of head water can only be lowered a 
comparatively small amount without impairing the output 
and efficiency of the plant. 

Water is led from the head-race or the reservoir through 



290 



TRACTION AND TRANSMISSION. 



suitable hand- or motor-operated head gates to the forebay 
and from there to the wheel pits. The water in entering 
the wheel pit from the head-race usually passes through a 
trash rack consisting of narrow iron bars, the function of 
which is to prevent large floating objects from entering the 
turbines. Open wheel pits are usual for heads up to 30 
feetj whereas closed flumes or penstocks leading from the 




Fig. 121. 



head-race to the wheel pits are utilized for higher heads. 
It is desirable to set the turbines in separate pits so that 
one or more may be temporarily shut down without inter- 
fering with the operation of the station. 

A cross-sectional view of the Rocky Creek Power House 
of the Southern Power Company is shown in Fig. 122, which 
also illustrates the construction of the penstock and draft 
tube for each turbine, and the water-tight stuffing box 
between the wheel pit and the generator room. 



HYDRAULIC STATIONS. 



291 




Fig. 122. 



292 



TRACTlt)N AND TRANSMISSION. 



Fig. 123 shows the interior of the Rainbow Station of the 
Great Falls Power Company, Montana. Each of the six 
3500 K.W. alternators is driven by a 6000 H.P. reaction 
turbine with two runners, each runner being enclosed 
in a separate spiral casing fed by a separate 8-foot steel 




Fig. 123. 

penstock from a balancing reservoir and discharging into 
a common draft tube. 

(2) High-head developments require long canals or pipe 
lines for conveying water from the intake to the power 
house. Level canals may be constructed along the hillside 
to a point above the power station, and from there the 
water can be passed down to the water wheels through a 



HYDRAULIC STATIONS. 293 

penstock. It is usually cheaper, however, to use a pipe 
line which need not be level but can follow the contour of 
the land. Wood, cast-iron, or riveted wrought-iron pipe is 
used for such purposes. The transmission of water through 
pipes or canals is accompanied by a reduction in the avail- 
able head, the extent of which depends upon the size of the 
pipe or canal. This loss of head can be computed from 
expressions given in most books on Hydraulics. 

Provision must be made to prevent injury to penstocks 
or pipe lines which might occur when the turbine gates or 
water-wheel nozzles are regulated too quickly. Automatic 
relief valves of sufficient area may be employed at the lower 
end of the pipe, or either standpipes or surge tanks may be 
used to alter the velocity of the water in the pipes. 

Fig. 1 24 gives a sectional view of a typical power house in 
which impulse wheels are installed. Speed regulation of 
the prime movers is accompHshed by deflecting the nozzles 
past the buckets and allowing part of the water to impinge 
upon heavy metal deflector plates. 

Frequently hydrauKc developments have auxiliary steam 
or gas engine plants to supplement the water power during 
the dry seasons or during periods of peak loads. 

100. Cost of Development. — The cost of a proposed 
hydraulic development depends largely upon the extent to 
w^hich the stream flow is to be developed, upon the nature 
and remoteness of the power market, as well as upon 
various topographical, geological, and meteorological con- 
ditions of the locality. The decision as to the commercial 
feasibility of a proposed water-power development must 
embrace a careful study of all such factors which influence 
water supply, of the available head and its variations, of 

the power available with and without pondage, of the 
20 



294 



TRACTION AND TRANSMISSION. 



/S^IO'A 00051 




fflK. 



fcr 



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'A. I f L t^V■..v^.■;.,v■■r.^-.■■■y^J.-^ 



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-^97X8' 



t<mmM 




HYDRAULIC STATIONS. 



^95 



location and extent of the hydraulic construction and 
power house, of the probable market for the power gener- 
ated and its load factor, and the desirability of auxiliary 
power. 

Rough estimates in terms of generator capacity of the 
cost of turbine equipments may be derived from Figs. 125 
and 126, which embody data from existing installations. 



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20 40 60 80 100 120 

HEAD iN FEET. 

Fig. 125. 

The figures refer to reaction turbines and impulse wheels 
respectively, and include extra movers for exciter units, 
governors, and cost of erection. 

The following table, given by 0. S. Lyford, gives the item- 
ized cost (estimated or actual) per kilowatt of generator 
capacity of seven separate water-power developments in 
the same general district in our southeastern states, these 
powers being developed with heads varying from 30 to 120 
feet, and with generator capacity varying from 10,000 to 



296 



TRACTION AND TRANSMISSION. 



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including righ 
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HYDRAULIC STATIONS. 



297 



30,cx)o K.W. The appended column gives the average of 
the proportional costs of the general groups for the seven 
plants. 



cc 

LJ 
Q. 

CD 
CC 
< 

_J 
-J 
O 
Q 



16 




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400 800 1200 

HEAD IN FEET. 

Fig. 126. 



1600 



loi. Depreciation and Obsolescence. — It is difficult to 
predetermine with accuracy the cost of repairs essential to 
maintain the various parts of an installation in operating 
condition, the time that these parts will endure before it 
becomes unwise to repair them, and the time which will 
elapse before it will prove more economical to substitute 
for them more efficient parts. It is necessary, however, to 
attempt to make such predeterminations in order to carry 
out an economic design. The following values in reference 
to hydraulic plants are those given by Dr. Cary T. Hutch- 
inson. He also states that the general consensus of opinion 
as to the depreciation of steam generating plants is that it 
amounts to from 5 per cent to 7.5 per cent, with an addi- 
tional like value for obsolescence. The basis of the follow- 



298 



TRACTION AND TRANSMISSION- 



ing table is the assumed life and annual charges compounded 
at the rate of 4.5 per cent. The depreciation in terms of 
the total cost assumes that the cost of the power house, 
the transmission line, and the substation amounts to but 
57 per cent of the total cost. 



DEPRECIATION RATES. 



Item. 



Power House: 

1. Stop logs, gates, and other wood 

exposed to air and water 

2. Flooring, roofing and hardware, 

and miscellaneous fixtures .... 

3. Tile wainscoting, sewage, plumb- 

ing system, and metal window 
frames, etc 

4. Electric light and telephone. . . . 

5. Switchboard equipment 

6. Cables and heavy wiring 

7. Cranes 

8. Water wheels 

o. Water-wheel governors 

10. Generators and transformers. . 



Transmission Line: 

1. Right of way 

2. Towers 

3. Special structures. 

4. Insulators 

5. Copper 

6. Installation 



Substation : 



Land 

Buildings 

Transformers . 
Switches, etc. 
Installation. . . 



Propor- 
tional cost. 



0.80 



9.80 



2.45 
0.80 



4 
3 
I 

33 

2. 



35 
90 

25 
75 
90 



40.00 



45. 
18. 

5- 
2. 

23- 
5. 



100. o 

6.0 
30. 
40. 
16. 

8. 
100. 



Life 
years. 



[19I 

5 

15 



15 
10 
10 
10 
15 
25 
10 

25 



[26] 



15 
10 
10 
25 



[20] 



25 
20 
10 



Annual amount 

for depreciation 

in per cent of 

total cost. 



0.146 
0.472 



O.I18 
0.065 

0.355 
0.318 
0.060 

0.757 
0.235 
0.898 

3 423 



0.885 

0.415 
o. 170 

0.530 



0.67 
1.28 
1 .29 



3 24 



HYDRAULIC STATIONS. 



299 



102. Relative Operating Expenses. — The following 
table, due to H. G. Stott, is applicable to plants having a 
maximum load of over 30,000 K.W., and gives operating 
expenses and probable fixed charges based upon 5 per cent 
interest, i per cent for taxes and general administrative 
expenses, and 5 per cent amortization or obsolescence in 
the steam and hydraulic plants. 

RELATIVE COSTS PER KILOWATT-HOUR. 



10. 
II. 

12. 

13- 
14. 



Items. 



Maintenance 
Engine room, mechanical. . . . 

Boiler or producer room 

Coal-and ash-handling apparatus 
Electrical apparatus 



Operation 

Coal 

Water 

Engine-room labor 

Boiler- or producer-room labor 
Coal- and ash-handling labor . 

Ash removal 

Electrical labor 

Engine-room lubrication 

Engine-room waste, etc 

Boiler-room lubrication, etc.. . 



Relative operating cost, per cent. . 
Relative investment, per cent. . . . 
Probable average cost, per K.W.($; 
Probable fixed charges 



2.59 
4-65 
0.58 
I -13 



61 

7 
6 

7 
2 
I 
2, 
I , 
o, 
o, 



100.00 
100.00 
125.00 

11% 



0.51 
4.33 
0.54 
I -13 



55-53 
0.65 
1.36 
6.74 
2.13 
0.95 
2.54 
0.35 
0.30 
0.17 



77-23 
75.00 

93-75 
11% 



I fe: '^ 

« n ^ 

■■+J a 

03 OS p 






1-55 
3-55 
0.44 

I -13 



52.44 
0.61 



0.30 
0.17 



75-87 

80.00 

100.00 

11% 



5-18 
1. 16 
o. 29 

1. 13 

26.52 
3.60 
6.76 
1. 81 

1. 14 
o. 

2. 
I . 
O. 
O. 



52.94 
110.00 

137-50 

12% 



CD 03 



2.84 

1-97 
0.29 

I -13 



25-97 
2.16 



47.23 

96.20 

120.00 

11.5% 



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2.54 
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0.20 



5-94 
100.00 
125.00 
11% 



103. Costs per Kilowatt-hour. — The average annual 
cost per kilowatt-hour of output depends upon the annual 



300 



TRACTION AND TRANSMISSION. 



load factor and upon the type of an installation. The annual 
load factor is the ratio of the annual output in kilowatt- 
hours to 8760 times the maximum power of the installed 
apparatus in kilowatts. Since the fixed charges are de- 
pendent upon the rated capacity but independent of the 

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LOAD FACTOR. 

Fig. 127. 



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output, whereas the operating expenses are dependent upon 
the latter and independent of the former, the cost per kilo- 
watt-hour of output will be a minimum for a load factor of 
unity. Furthermore, for a typical railway load of a given 
maximum demand the rating of the power-station equip- 
ment necessarily installed to meet this demand differs with 
the type of the installation. This is due to differences in 



HYDRAULIC STATIONS. 301 

overload capacity. The necessary capacity progressively 
increases as the type changes from steam to gas and steam 
again to hydraulic or to gas alone. 

For a complete discussion of this subject the reader is 
referred to Mr. Stott's paper (Trans. A. I. E. E., xxviii, 
p. 1479), from which Fig. 127 is taken. This figure shows 
the dependence of the total annual cost per installed 
kilowatt upon the load factor and the type of plant. The 
titles associated with the various lines refer to the col- 
umns in the table of the preceding article, each of which 
represents a definite type of installation. A low grade of 
coal, costing ^1.50 per ton and giving 11,000 B.t.u. per 
pound, has been assumed. The average cost per kilowatt- 
hour may be determined by dividing the value of any 
ordinate by 8760 times the corresponding load-factor. 

PROBLEMS. 

48. Determine the proper size and number of steam turbo-generator units 
for a power station having a load curve of the form mdicated in Fig. 107 
but with ordinates of half the value. What would be the probable number 
of daily hours of operation of each unit ? 

49. If the turbines of problem 48 consume 17 pounds of dry saturated 
steam at 175 pounds gauge pressure per kilowatt-hour and if the auxiliaries 
use 10 per cent of the total steam generated, how many boilers should be 
installed per unit and what should be the horsepower of each ? Assume 
the temperature of feed water to be 80° F. 

50. Determine the diameter of the runners for a twin reaction turbine to 
operate on a 100-foot head for a 5000-kilowatt, 25-cycle, three-phase alter- 
nator, whose efficiency is 96 per cent. The constants of the turbine are 

r = 0.74 

€ = 0.85 



INDEX. 



Acceleration, 22. 
automatic, 108. 
curve, S3, 57. 
rates, changes in, 126. 
Adequacy of copper distribution, 

150. 
Adhesion, coefficient of, 50. 
Adjustment of speed curves, (yG. 
Alternating-current control, 90. 
distribution, 164. 
motors, 28. 
substations, 166. 
■ Aluminum-cell arrester, 256. 
Annual car-miles operated, 5. 
Apparatus, arrangement of station, 

189, 275. 
Arc suppressor, 257. 
Arresters, lightning, 209, 254. 
Atmospheric heaters, 272. 

potential diiferences, 252. 
Attenuation constant, 238. 
Automatic acceleration, 108. 

substations, 188. 
Auxiliary feeders, 151. 

storage batteries, 188. 
Average current per car, 112. 

Batteries, storage, 188. 
Bearing friction, 15. 
Boilers, 270. 
Bonds, track, 156. 
Boosters, 152. 
Braking, 23. 

curve, 52, 58. 

energy lost in, 126. 
Branches in roadway, 139. 

Cables, resistance of, 221. 
Capacity of lines, 230. 

of motors, 51. 
Car-body, types of, 9. 

cross sections, 17. 

equipments, weights of, 55. 



Car-mile, earnings per, 6. 

-miles, annual, 5. 

number of, for urban road, 4. 

propulsion, tractive effort for, 15. 

size of, 8. 

types of, 9. 
Cascade control, 100. 
Center feeding of sections, 138. 

of distribution, 199. 
Charging current of line, 247. 
Chimneys, 272. 
Choke coils, 209, 254. 
Classification of conductors, 133. 
Closed cars, 9. 
Coasting curve, 52, 58. 

effect of changes in, 129. 
Coefficient of adhesion, 24, 50. 
Collecting devices, 140. 
Commutating-pole motors, 27, 55. 
Compensated series motors, 36. 
Compensators, 90, 93. 

multiple-switch, 94. 
Compounded converters, 172. 
Condensers, 267. 
Conductive compensation, 37. 
Conductor separation, 213. 
Conductors, resistance of, 220. 

weights of, 202, 220. 
Connecting-rod drive, 41. 
Contact conductors, 134. 
Continuous rating of motors, 119. 
Control, alternating-current, 90. 

apparatus, weights of, 55. 

cascade, 100. 

compensator, 93. 

direct-current, 74. 

field excitation, 89. 

hand, 103. 

induction motor, 96. 
regulator, 90. 

methods of, 74. 

multiple-unit, 105. 

rheostatic, 74. 

series-parallel, 75. 



303 



304 



INDEX. 



Controllers, 103. 

Converter, characteristics of, 171. 

substations, 169. 

-transformer deficlences, 183. 
Convertible cars, 9. 
Cooling towers, 269. 
Copper loss of motor, 118. 
Corona, 213. 

loss, 214, 247. 
Corrosion, electrolytic, 157. 
Cost constants, 185. 

of electrical energy, 299. 

of hydraulic development, 293. 
movers, 295. 

of steam stations, 279. 

of substation units, 174. 

of transformers, 208. 
Critical line voltage, 213. 
Cross section of contact conductor, 

135- 
of feeder, 151. 
of line conductor, 206. 
of supplementary conductor, 

143. 
Current, average, per car, 112. 

curves, 1 11. 

density, economic, 152. 

distribution on lines, 240. 

eifective, motor, 113. 

-limit relay, 109. 
Curves in roadway, 20. 

Daily load diagrams, 180. 

Dams, 288. 

Data for plotting speed curves, 53. 

Deficiency constants, 184. 

Degree of track curvature, 21. 

Density factor of air, 215. 

Depreciation of generating plants, 

297. 
Design of controller units, 79. 
Developments, hydraulic, 288. 

cost of, 293. 
Direct-current control, 74. 

motors, 27. 

transmission, 166. 
Disruptive critical voltage, 219. 
Distance curves, 66. 
Distributing system, 133. 
Distribution of current on lines, 240. 
Diversity factor, 210, 260. 



Double-decked cars, 9. 

stations, 275. 
Drive, methods of, 41. 
Duration of stops, 56. 

Earnings per car-mile, 5. 
Economic current density, 152. 

section of contact conductor, 135, 
177. 

spacing of substations, 176. 

transmission voltage, 205. 
Economizers, 272. 
Effective motor current, 113. 

per trip, 116. 
Effect of operating conditions on 

energy consumption, 124. 
Efficiency of hydraulic movers, 285. 

of substation apparatus, 170. 

of transformers, 168, 203. 

of transmission, 246. 
Electrical energy, cost of, 299. 
Electric field intensity near con- 
ductors, 214. 
Electrolytic corrosion, 157. 

surveys, 161. 
E.M.F. equation of single-phase 

motors, 32, 38. 
Elevation of outer rail, 22. 
End feeding of sections, 137. 
Energy consumption, 1 11. 

effect of operation on, 124. 

for car propulsion, 120. 
Engineer's problem, I. 
Engines, steam, 265. 
Equations of wave propagation, 

235- 
Equivalent grade, 20. 

hours of operation, 182. 

line length, 211. 
Expenses per car-mile, 6. 

Feeders, 151. 

negative, 157. 
Feed-water heaters, 272. 
Field control, 89. 

Fixed charges of power station, 264. 
Floor space in power stations, 274. 

in substations, 170. 
Forced compensation, 37. 
Frequency, 203. 

resonant, of line, 204. 



INDEX. 



305 



Friction, coeiBcIent of, 24. 

Gas engines, 264. 
Gates for turbines, 282. 
Gear drive, 41. 

ratio, choice of, 56. 

effect on acceleration rate, 131. 
Generators for power station, 261. 
Governors for hydraulic movers, 284. 
Grades, 20. 

Graphic time-tables, 147. 
Grid resistances, 80. 
Ground wires, 258. 

Hand control, 103. 
Heating of motors, 51, 118. 
Heights of chimneys, 273. 
Horsepower rating of motors, 119. 
Hydraulic construction, 288. 

power stations, 281. 
Hyperbolic functions, 224. 

Impedance of rails, 164. 
Impulse wheels, 281. 
Income of electric railways, 5. 
Inductance of lines, 222. 
Induction motor, 40. 
control of, 96. 
regulators, 90. 
Inductive compensation, 37. 
Ingredients of third rails, 130. 
Installations, substation, cost of, 

175. 
Insulators, 207. 

Internal combustion engines, 264. 
Interpoie motors, 27, 55. 
Ionization of air, 215. 
Iron loss of motor, 118. 

pipes, resistance of, 163. 

Jet condensers, 268. 

Leakage current, 157. 

Leakance, line, 237. 

Length of average passenger ride, 8. 

of track for urban road, 2. 
Lightning, 251. 

arresters, 209, 254. 

protection, 254. 
Limitations of motors, 50. 
Limiting line voltages, 219. 



Line capacity, 230. 

inductance, 222. 

leakance, 237. 

resistance, 220. 
Load curves, 180, 259. 
Location of substations, 175. 

of transmission line, 199. 
Locomotives, electric, 40. 
Losses in motors, 118. 

in substations, 184. 

Master controllers, 106. 
Mechanical draft apparatus, 273. 
Mixed-flow turbine, 281. 
Motor capacity, 51. 

characteristic curves, 44. 

control, 74. 

effective current, 113. 

-generator substations, 170. 

heating, 51, 118. 

limitations, 50. 

output, 49. 

saturation curve, 79, 87. 
Motors, alternating-current, 26. 

compensated series, 36. 

direct-current, 27. 

field-control, 89. 

induction, 40. 

railway, 26. 

repulsion, 39. 

series, 27, 30. 

weights of, 55. 
Moutiers-Lyons transmission, 166. 
Multiple-switch compensator, 94. 

-unit control, 105. 

Narragansett type of car, 10. 
Negative conductors, 133. 

track feeders, 157. 
Nominal rating of motors, no. 
Number of cars for urban road, 4. 

of units in substations, 180. 
Numerical examples, 18, 59, G'jy 
87, 113, 127, 186, 205, 211, 
218, 244, 250, 287. 

Obsolescence of generating plants, 

297. 
Oil switches, cost of, 209. 
One-man cars, 10. 
Open cars, 9. 



3o6 



INDEX. 



Operating characteristics of con- 
verters, 171. 
of motor-generators, 172. 
conditions, changes in, 124. 
expenses of power stations, 264, 
280, 299. 
of railways, 6. 
Output of power stations, 261. 
Overload capacity of generators, 182, 
263. 
coefficient, 180. 
Overrunning third rail, 141. 
Oxide-film arrester, 256. 

Pantograph frames, 141. 
Passenger factor, 3. 
Pay-as-you-enter cars, 10. 
Performance curves of motors, 44. 
Phase converters, 29. 
Phases, number of, 201. 
Pipes, resistances of, 163. 
Plotting speed curves, 56. 

with grades and curves, G'f. 
Polarity induction motor control, 97. 
Poles, transmission, 207. 

trolley, 140. 
Population served by railway, 4. 
Portable substations, 193. 
Positive conductors, 133. 
Power factor curves, 122. 

of single-phase motors, 34, 38. 
lost in conductors, 138. 
-station buildings, 274. 
costs, 264, 279. 
location of, 200. 
output, 261. 
Preventive coils, 93. 
Prime movers, types of, 263. 
Problems, 14, 25, 49, 73, no, 132, 

164, 197, 258, 301. 
Propagation of electric waves, 235. 
Protection from lightning, 254. 
Pumps for steam stations, 268. 

Quill drive, 43. 

Rails, 155. 

impedance of, 164. 
Rates of acceleration, 23. 

of braking, 53. 



Reactance set, 172. 
Receipts of electric railway, 4. 
Regeneration of energy with induc- 
tion motors, 40. 
Regulation of converters, 171. 

of transmission line, 243. 
Regulators, induction, 90. 
Relative operating expenses of gen- 
erating plants, 299. 

weights of conductors, 202. 
Relay, current-limit, 109. 
Repulsion motors, 39. 
Resistance of conductors, 220. 

offered to car movement, 15. 

of iron pipes, 163. 

of third rails, 136. 

of track rails, 156. 

to alternating currents, 221. 
Resistances, motor starting, 78. 
Resonant currents, 253. 

frequency of line, 204. 
Retardation, 24. 
Reversing motors, 103. 
Rheostatic control, 74. 
Ride, average passenger, 8. 
Rights of way, 200. 
Roadway, characteristics of, 56. 
Rolling friction, 15. 

Saturation curve of motors, 79, 87. 

Schedule speeds, 13, 56. 

Scott transformer connection, 29, 

169. 
Seating capacity of cars, 9. 
Seats, arrangement of, 10. 
Sectional contact conductors, 138. 
Selection of gear ratio, 56. 

of generator units, 261. 
Semiconvertible cars, 9. 
Separation of line conductors, 213. 
Series-parallel control, 75. 

-wound motors, 27, 30. 
Service, railway, types of, I. 
Single-phase railway motors, 30. 
Skin effect, 221. 

resistance of rails, 164. 
Speed curves, 50. 

of car, 51. 

of hydraulic movers, 287. 

of motor, 26, 49. 



INDEX. 



307 



Stacks, 272. 

Standard transmission voltages, 212. 
Starting resistances, 78. 
energy lost in, 125. 
Station load curves, 259. 
Steam power stations, 265. 
Stops, duration of, 56. 
Storage batteries, 188. 
Substations, 166. 

arrangemicnt of apparatus in, 189. 

automatic, 188. 

cost of, 175. 

efficiency of apparatus in, 170. 

floor space in, 170. 

location of, 175. 

number of units in, 180. 

portable, 193. 

connections of, 197. 
Superheaters, 266. 
Supplementary conductors, 142. 
Surface condensers, 268. 
Surges from lightning, 253. 
Surveys, electrolytic, 161. 
Synchronous speed of induction 
motors, 97. 

Table of hyperbolic functions, 228. 
Temperature elevation of motors, 

Third rails, composition of, 136. 

resistance of, 136. 
Three-phase railway motors, 40. 

-point grid resistance, 80. 
Thury transmission system, 166. 
Time-tables, graphic, 148. 
Total drop in conductor, 135. 
Towers, transmission, 207. 
Track factor, 3. 

feeders, 157. 

length of, for urban road, 2. 

rails, 155. 
Traction motors, 26. 
Tractive effort, 15, 49. 
-speed curve, 60, 



Train resistance, 15. 

-sheets, 148. 
Trains, 13. 

Transformer efficiencies, 168, 203. 
Transformers, costs of, 208. 

weights of, 203. 
Transmission lines, 199. 
Trolley wheels, 140. 

wires, 135. 
Turbines, hydraulic, 281. 

steam, 265. 
Typical speed curves, 52. 

Underrunning third rail, 142. 
Units, controller resistance, 79. 

generator, 262. 
Urban road, cars for, i. 

Vacuum heaters, 272. 

pumps, 268. 
Velocity of car, 51. 

of wave propagation, 243. 
Voltage along roadway, 129. 

critical, 213. 

curves, 118. 

distribution on lines, 240. 

gradient, 214. 

of boosters, 154. 

regulation, 243. 

transmission, economic, 205. 

Wages of substation attendants, 178. 
Water-power development, 288. 

wheels, 284. 
Watts lost in conductor, 138. 
Wave-length coefficient, 238. 

propagation along wires, 235. 
Weights of car equipments, 54. 

of cars, 13. 

of conductors, relative, 202. 

of iron pipe, 163. 

of transformers, 203. 
Wheels, trolley, 140. 
Wind resistance, 16. 



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